156:
6067:
1795:
visual art to convey the artist's message and to express certain ideas. However, differences in worldview (beliefs and culture) had a great impact on artists' perceptions. While some emphasised the circle's perimeter to demonstrate their democratic manifestation, others focused on its centre to symbolise the concept of cosmic unity. In mystical doctrines, the circle mainly symbolises the infinite and cyclical nature of existence, but in religious traditions it represents heavenly bodies and divine spirits.
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1610:
3470:{\displaystyle {\frac {({\color {green}x}-x_{1})({\color {green}x}-x_{2})+({\color {red}y}-y_{1})({\color {red}y}-y_{2})}{({\color {red}y}-y_{1})({\color {green}x}-x_{2})-({\color {red}y}-y_{2})({\color {green}x}-x_{1})}}={\frac {(x_{3}-x_{1})(x_{3}-x_{2})+(y_{3}-y_{1})(y_{3}-y_{2})}{(y_{3}-y_{1})(x_{3}-x_{2})-(y_{3}-y_{2})(x_{3}-x_{1})}}.}
1502:: one of the two possible arcs determined by the endpoints of a diameter, taking its midpoint as centre. In non-technical common usage it may mean the interior of the two-dimensional region bounded by a diameter and one of its arcs, that is technically called a half-disc. A half-disc is a special case of a segment, namely the largest one.
1406:: a line segment whose endpoints lie on the circle and that passes through the centre; or the length of such a line segment. This is the largest distance between any two points on the circle. It is a special case of a chord, namely the longest chord for a given circle, and its length is twice the length of a radius.
1798:
The circle signifies many sacred and spiritual concepts, including unity, infinity, wholeness, the universe, divinity, balance, stability and perfection, among others. Such concepts have been conveyed in cultures worldwide through the use of symbols, for example, a compass, a halo, the vesica piscis
1794:
From the time of the earliest known civilisations – such as the
Assyrians and ancient Egyptians, those in the Indus Valley and along the Yellow River in China, and the Western civilisations of ancient Greece and Rome during classical Antiquity – the circle has been used directly or indirectly in
7322:
6625:
is the case in which the weights are equal. A circle is an ellipse with an eccentricity of zero, meaning that the two foci coincide with each other as the centre of the circle. A circle is also a different special case of a
Cartesian oval in which one of the weights is
6754:
is a figure whose width, defined as the perpendicular distance between two distinct parallel lines each intersecting its boundary in a single point, is the same regardless of the direction of those two parallel lines. The circle is the simplest example of this type of
5166:
Through any three points, not all on the same line, there lies a unique circle. In
Cartesian coordinates, it is possible to give explicit formulae for the coordinates of the centre of the circle and the radius in terms of the coordinates of the three given points. See
7458:
2834:
6784:
points in the plane. The locus of points such that the sum of the squares of the distances to the given points is constant is a circle, whose centre is at the centroid of the given points. A generalization for higher powers of distances is obtained if under
4025:
7023:
4429:
2652:
6709:
7660:
1691:
A circle is a plane figure bounded by one curved line, and such that all straight lines drawn from a certain point within it to the bounding line, are equal. The bounding line is called its circumference and the point, its
7174:
2074:
4709:
7331:
3721:
4920:
2696:
6563:, passing through each vertex. A well-studied example is the cyclic quadrilateral. Every regular polygon and every triangle is a cyclic polygon. A polygon that is both cyclic and tangential is called a
5754:(red). Hence, all inscribed angles that subtend the same arc (pink) are equal. Angles inscribed on the arc (brown) are supplementary. In particular, every inscribed angle that subtends a diameter is a
5322:
The sum of the squared lengths of any two chords intersecting at right angles at a given point is the same as that of any other two perpendicular chords intersecting at the same point and is given by 8
4322:
6481:
5061:
2978:
5721:
4208:
1424:: a line segment joining the centre of a circle with any single point on the circle itself; or the length of such a segment, which is half (the length of) a diameter. Usually, the radius is denoted
5852:
4819:
1412:: the region of the plane bounded by a circle. In strict mathematical usage, a circle is only the boundary of the disc (or disk), while in everyday use the term "circle" may also refer to a disc.
2701:
2569:
7059:, the loci of the constant sums of the second and fourth powers are circles, whereas for the square, the loci are circles for the constant sums of the second, fourth, and sixth powers. For the
6212:
5864:, since the sagitta intersects the midpoint of the chord, we know that it is a part of a diameter of the circle. Since the diameter is twice the radius, the "missing" part of the diameter is (
5558:
The angle between a chord and the tangent at one of its endpoints is equal to one half the angle subtended at the centre of the circle, on the opposite side of the chord (tangent chord angle).
3582:
5953:
Construct a circle through points A, B and C by finding the perpendicular bisectors (red) of the sides of the triangle (blue). Only two of the three bisectors are needed to find the centre.
2296:
1486:: a region bounded by a chord and one of the arcs connecting the chord's endpoints. The length of the chord imposes a lower boundary on the diameter of possible arcs. Sometimes the term
2458:
1993:
5653:
5621:
3899:
3910:
2161:
6905:
6349:
4991:
1720:
there is a detailed definition and explanation of the circle. Plato explains the perfect circle, and how it is different from any drawing, words, definition or explanation. Early
8645:
2544:
7694:
parallel to the coordinate axes, so planar
Chebyshev distance can be viewed as equivalent by rotation and scaling to planar taxicab distance. However, this equivalence between
1902:
8053:
3798:
2564:
7499:
6506:
satisfying the above equation is called a "generalised circle." It may either be a true circle or a line. In this sense a line is a generalised circle of infinite radius.
4140:
3759:
2229:
6747:
is a set of points such that the product of the distances from any of its points to two fixed points is a constant. When the two fixed points coincide, a circle results.
5875:) in length. Using the fact that one part of one chord times the other part is equal to the same product taken along a chord intersecting the first chord, we find that (
7592:
7587:
4090:
5341:
The distance from a point on the circle to a given chord times the diameter of the circle equals the product of the distances from the point to the ends of the chord.
7050:
6880:
6830:
7535:
4327:
1480:: a region bounded by two radii of equal length with a common centre and either of the two possible arcs, determined by this centre and the endpoints of the radii.
7164:
Defining a circle as the set of points with a fixed distance from a point, different shapes can be considered circles under different definitions of distance. In
1468:
8201:
6853:
6900:
6803:
6782:
2087:
The circle is the plane curve enclosing the maximum area for a given arc length. This relates the circle to a problem in the calculus of variations, namely the
2004:
1442:
2474:
applied to any point on the circle: as shown in the adjacent diagram, the radius is the hypotenuse of a right-angled triangle whose other sides are of length |
8623:
8123:
6636:
4571:
1284:
8421:
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is equal to that of a triangle whose base has the length of the circle's circumference and whose height equals the circle's radius, which comes to
7154:-norms (every vector from the origin to the unit circle has a length of one, the length being calculated with length-formula of the corresponding
6574:
is a curve that is inscribed in a given circle by tracing a fixed point on a smaller circle that rolls within and tangent to the given circle.
6386:
4996:
8311:
5908:
4147:
1733:
384:
7830:
6525:
About every triangle a unique circle, called the circumcircle, can be circumscribed such that it goes through each of the triangle's three
6519:
5796:
2862:
is made to range not only through all reals but also to a point at infinity; otherwise, the leftmost point of the circle would be omitted.
4714:
6154:
5188:
of a chord passes through the centre of a circle; equivalent statements stemming from the uniqueness of the perpendicular bisector are:
7317:{\displaystyle \left\|x\right\|_{p}=\left(\left|x_{1}\right|^{p}+\left|x_{2}\right|^{p}+\dotsb +\left|x_{n}\right|^{p}\right)^{1/p}.}
7092:
5914:
The simplest and most basic is the construction given the centre of the circle and a point on the circle. Place the fixed leg of the
5793:
of the sagitta, the
Pythagorean theorem can be used to calculate the radius of the unique circle that will fit around the two lines:
3495:
8705:
8278:
350:
5211:
of a circle are subtended by the same chord and on the same side of the chord, then the central angle is twice the inscribed angle.
2252:
1344:, makes much of modern machinery possible. In mathematics, the study of the circle has helped inspire the development of geometry,
8050:
8540:
8159:
1603:
4231:
2383:
1948:
8002:
2873:
2855:
3851:
8672:
5658:
2130:
1277:
1231:
837:
296:
105:
8521:
8182:
7453:{\displaystyle \left\|x\right\|_{2}={\sqrt {\left|x_{1}\right|^{2}+\left|x_{2}\right|^{2}+\dotsb +\left|x_{n}\right|^{2}}}.}
5856:
Another proof of this result, which relies only on two chord properties given above, is as follows. Given a chord of length
8075:
6290:
5354:
A line drawn perpendicular to a tangent through the point of contact with a circle passes through the centre of the circle.
1378:
part of a circle. Specifying two end points of an arc and a centre allows for two arcs that together make up a full circle.
8732:
5351:
A line drawn perpendicular to a radius through the end point of the radius lying on the circle is a tangent to the circle.
5116:
4934:
7987:
8502:
2493:
1787:
8120:
5521:
A tangent can be considered a limiting case of a secant whose ends are coincident. If a tangent from an external point
5357:
Two tangents can always be drawn to a circle from any point outside the circle, and these tangents are equal in length.
8727:
8393:
8198:
8027:
6583:
5253:
3607:
2353:
7768:
2829:{\displaystyle {\begin{aligned}x&=a+r{\frac {1-t^{2}}{1+t^{2}}},\\y&=b+r{\frac {2t}{1+t^{2}}}.\end{aligned}}}
1863:
3815: axis to the line connecting the origin to the centre of the circle). For a circle centred on the origin, i.e.
8817:
6357:
is a point on the circle of
Apollonius if and only if the cross-ratio is on the unit circle in the complex plane.
1252:
862:
5782:) is a line segment drawn perpendicular to a chord, between the midpoint of that chord and the arc of the circle.
4097:
1508:: a coplanar straight line that has one single point in common with a circle ("touches the circle at this point").
1332:
The circle has been known since before the beginning of recorded history. Natural circles are common, such as the
2202:
1736:, and many believed that there was something intrinsically "divine" or "perfect" that could be found in circles.
1270:
8593:
1321:. The length of a line segment connecting two points on the circle and passing through the centre is called the
8780:
7788:
7666:
7085:
6537:
5072:
4925:
4444:
3587:
2847:
2088:
1567:
1471:
239:
5626:
5594:
4050:
8822:
8457:
7885:
7084:, of constructing a square with the same area as a given circle by using only a finite number of steps with
6751:
5750:
An inscribed angle (examples are the blue and green angles in the figure) is exactly half the corresponding
5737:
5247:
Among all the circles with a chord AB in common, the circle with minimal radius is the one with diameter AB.
3485:
1934:
1740:
1729:
665:
345:
202:
31:
7875:
6607:
6137:
6000:
5985:
5915:
5185:
2558:
1598:
741:
452:
330:
215:
2307:, these formulae yield the circumference of a complete circle and area of a complete disc, respectively.
8639:
8244:
7992:
7982:
7735:
7100:
6066:
6061:
5109:
4431:. Not all generalised circles are actually circles: a generalised circle is either a (true) circle or a
1838:
1748:
1361:
513:
474:
433:
428:
281:
7734:. Two topological circles are equivalent if one can be transformed into the other via a deformation of
8269:
Abdullahi, Yahya (29 October 2019). "The Circle from East to West". In
Charnier, Jean-François (ed.).
1130:
877:
8433:
7977:
7056:
5225:
5218:
5214:
If two angles are inscribed on the same chord and on the same side of the chord, then they are equal.
1705:
1701:
1681:
1181:
1104:
952:
857:
379:
274:
188:
5945:
passing through one of the endpoints of the diameter (it will also pass through the other endpoint).
8007:
7078:
7072:
6499:
6090:
5083:
5079:
3764:
2659:
2554:
2471:
2169:
the angle at the centre of the circle. The angle subtended by a complete circle at its centre is a
1808:
1752:
1310:
1186:
1043:
897:
802:
692:
563:
553:
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291:
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232:
184:
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160:
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155:
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8103:
7942:
7912:
7763:
7758:
7677:
7108:
6533:
6366:
5775:
4225:
3732:
3586:
It can be proven that a conic section is a circle exactly when it contains (when extended to the
1779:
1409:
1326:
1145:
872:
712:
340:
264:
254:
225:
210:
5238:
5217:
If two angles are inscribed on the same chord and on opposite sides of the chord, then they are
4020:{\displaystyle r=r_{0}\cos(\theta -\phi )\pm {\sqrt {a^{2}-r_{0}^{2}\sin ^{2}(\theta -\phi )}}.}
8791:
7540:
6283:
are as above, then the circle of
Apollonius for these three points is the collection of points
8751:
8701:
8668:
8470:
8307:
8303:
8296:
8274:
8095:
7895:
7870:
7018:{\displaystyle \sum _{i=1}^{n}d_{i}^{2m}>nR^{2m},\quad {\text{ where }}~m=1,2,\dots ,n-1;}
6564:
6526:
5158:
4931:
When the centre of the circle is at the origin, then the equation of the tangent line becomes
3619:
2357:
2104:
1835:
1763:
1381:
1314:
1216:
1206:
1135:
1004:
982:
932:
907:
842:
766:
421:
313:
259:
220:
7474:
with sides oriented at a 45° angle to the coordinate axes. While each side would have length
4424:{\displaystyle |z-c|^{2}=z{\overline {z}}-{\overline {c}}z-c{\overline {z}}+c{\overline {c}}}
1390:: a line segment whose endpoints lie on the circle, thus dividing a circle into two segments.
8812:
8722:
8615:
8566:
8441:
8354:
8087:
7930:
7860:
7783:
7502:
7463:
7104:
7060:
6545:
6259:
Second, see for a proof that every point on the indicated circle satisfies the given ratio.
6105:. (The set of points where the distances are equal is the perpendicular bisector of segment
5587:
If two secants are inscribed in the circle as shown at right, then the measurement of angle
5445:
5091:
2320:
1924:
1762:
in the early 20th century, geometric objects became an artistic subject in their own right.
1655:, dated to 1700 BCE, gives a method to find the area of a circle. The result corresponds to
1490:
is used only for regions not containing the centre of the circle to which their arc belongs.
1483:
1387:
1306:
1196:
937:
647:
525:
460:
318:
303:
168:
110:
8366:
7028:
6858:
6808:
6093:
showed that a circle may also be defined as the set of points in a plane having a constant
5181:
Chords are equidistant from the centre of a circle if and only if they are equal in length.
1841:
approximately equal to 3.141592654. The ratio of a circle's circumference to its radius is
8544:
8525:
8506:
8362:
8205:
8186:
8154:
8139:
8127:
8057:
7925:
7803:
7798:
7741:
7717:
7520:
7116:
6591:
6549:
5208:
5123:
4432:
2238:
1477:
1415:
1375:
619:
492:
482:
325:
308:
249:
1447:
1191:
1160:
1094:
1064:
942:
887:
882:
822:
8437:
8076:"Sacred landscapes of the south-eastern USA: prehistoric rock and cave art in Tennessee"
8074:
Simek, Jan F.; Cressler, Alan; Herrmann, Nicholas P.; Sherwood, Sarah C. (1 June 2013).
6835:
5918:
on the centre point, the movable leg on the point on the circle and rotate the compass.
5742:
8694:
8557:
Apostol, Tom; Mnatsakanian, Mamikon (2003). "Sums of squares of distances in m-space".
7972:
7890:
7865:
7850:
7835:
7808:
7778:
7773:
6885:
6788:
6767:
6614:
6556:
6541:
6522:, can be inscribed such that it is tangent to each of the three sides of the triangle.
6145:
5562:
5229:
5087:
2490:|. If the circle is centred at the origin (0, 0), then the equation simplifies to
2181:
2170:
1938:
1717:
1641:
1559:
1427:
1247:
1221:
1155:
1099:
972:
852:
832:
812:
717:
8537:
5436:
5078:
The circle is a highly symmetric shape: every line through the centre forms a line of
2647:{\displaystyle {\begin{aligned}x&=a+r\,\cos t,\\y&=b+r\,\sin t,\end{aligned}}}
8806:
8661:
8107:
7962:
7855:
7731:
6704:{\displaystyle \left|{\frac {x}{a}}\right|^{n}\!+\left|{\frac {y}{b}}\right|^{n}\!=1}
6621:
of the distances from any of its points to two fixed points (foci) is a constant. An
5751:
5199:
4036:
3489:
2246:
1825:
1783:
1652:
1637:
1629:
1571:
1393:
1226:
1211:
1140:
957:
917:
867:
642:
605:
572:
410:
406:
98:
8578:
8358:
7119:
coefficients. Despite the impossibility, this topic continues to be of interest for
8795:
8570:
8453:
7880:
7145:
7140:
7063:
the constant sum of the eighth powers of the distances will be added and so forth.
6744:
6630:
6618:
6560:
5285:
If the intersection of any two perpendicular chords divides one chord into lengths
5195:
5168:
5150:
5099:
5071:
The circle is the shape with the largest area for a given length of perimeter (see
4450:
2185:
2112:
2077:
1804:
1800:
1759:
1625:
1617:
1496:: an extended chord, a coplanar straight line, intersecting a circle in two points.
1371:
1165:
1114:
927:
782:
697:
487:
48:
8769:
8218:
7655:{\displaystyle r={\frac {1}{\left|\sin \theta \right|+\left|\cos \theta \right|}}}
4027:
Without the ± sign, the equation would in some cases describe only half a circle.
8518:
8177:
8827:
7813:
7793:
7720:
6571:
6268:
5949:
5755:
5154:
5143:
1493:
1201:
1074:
892:
827:
755:
727:
702:
6240:
extended. Since the interior and exterior angles sum to 180 degrees, the angle
1774:
1537:
8743:
8091:
7537:
is 4 in this geometry. The formula for the unit circle in taxicab geometry is
7120:
7112:
6603:
5591:
is equal to one half the difference of the measurements of the enclosed arcs (
2116:
1930:
1633:
1527:
1499:
1365:
1059:
1038:
1028:
1018:
977:
922:
817:
807:
707:
558:
30:
This article is about the shape and mathematical concept. For other uses, see
8499:
8099:
1799:
and its derivatives (fish, eye, aureole, mandorla, etc.), the ouroboros, the
8759:
8754:
8619:
7967:
7081:
6021:
5766:
1645:
1345:
1333:
1317:. The distance between any point of the circle and the centre is called the
1069:
787:
750:
614:
586:
131:
8143:
8179:
The Six Books of
Proclus, the Platonic Successor, on the Theology of Plato
2069:{\displaystyle \mathrm {Area} ={\frac {\pi d^{2}}{4}}\approx 0.7854d^{2},}
1687:
deals with the properties of circles. Euclid's definition of a circle is:
1586:
7845:
7840:
7727:
7681:
6515:
5928:
2463:
1725:
1403:
1349:
1322:
1150:
1109:
1079:
967:
962:
912:
637:
596:
544:
438:
401:
147:
6128:
satisfying the ratio of distances must fall on a particular circle. Let
1911:
17:
8173:
7091:
In 1882, the task was proven to be impossible, as a consequence of the
6622:
5779:
2981:
2166:
1721:
1505:
1084:
797:
591:
535:
335:
7730:, a circle is not limited to the geometric concept, but to all of its
6116:
The proof is in two parts. First, one must prove that, given two foci
2127:
measure 𝜃 of the angle is the ratio of the arc length to the radius:
1609:
8445:
7997:
7471:
7165:
6014:
Label the point of intersection of these two perpendicular bisectors
5142:
The circle that is centred at the origin with radius 1 is called the
2124:
1697:
1614:
1580:
1421:
1397:
1318:
1033:
1023:
902:
847:
722:
685:
673:
581:
499:
164:
8774:
for the properties of and elementary constructions involving circles
4704:{\displaystyle (x_{1}-a)x+(y_{1}-b)y=(x_{1}-a)x_{1}+(y_{1}-b)y_{1},}
1766:
in particular often used circles as an element of his compositions.
8610:
8327:
Richeson, David (2015). "Circular reasoning: who first proved that
6267:
A closely related property of circles involves the geometry of the
5338:
is the distance from the centre point to the point of intersection.
5191:
A perpendicular line from the centre of a circle bisects the chord.
1418:: the region common to (the intersection of) two overlapping discs.
1400:
of one circuit along the circle, or the distance around the circle.
8349:
7139:
6065:
5948:
5765:
5741:
5435:
2667:
2319:
2108:
1910:
1773:
1713:
1608:
1597:
1536:
1526:
1337:
1302:
1089:
1013:
947:
792:
396:
391:
6287:
for which the absolute value of the cross-ratio is equal to one:
6244:
is exactly 90 degrees; that is, a right angle. The set of points
6132:
be another point, also satisfying the ratio and lying on segment
5237:
An inscribed angle subtended by a diameter is a right angle (see
3807:
is the distance from the origin to the centre of the circle, and
3716:{\displaystyle r^{2}-2rr_{0}\cos(\theta -\phi )+r_{0}^{2}=a^{2},}
8739:
3761:
are the polar coordinates of a generic point on the circle, and
1341:
680:
530:
119:
4457:
on the circle is perpendicular to the diameter passing through
8547:. Mathworld.wolfram.com (2012-04-26). Retrieved on 2012-05-03.
8528:. Mathworld.wolfram.com (2012-04-26). Retrieved on 2012-05-03.
8509:. Mathworld.wolfram.com (2012-04-26). Retrieved on 2012-05-03.
3848:, or when the origin lies on the circle, the equation becomes
8785:
Click and drag points to see standard form equation in action
6882:
to the vertices of a given regular polygon with circumradius
4915:{\displaystyle {\frac {dy}{dx}}=-{\frac {x_{1}-a}{y_{1}-b}}.}
3800:
are the polar coordinates of the centre of the circle (i.e.,
7752:
1782:
in this 13th-century manuscript is a symbol of God's act of
7509:
is the circle's radius, its length in taxicab geometry is 2
4491:, then the tangent line is perpendicular to the line from (
1672:
1336:
or a slice of round fruit. The circle is the basis for the
4317:{\displaystyle p=1,\ g=-{\overline {c}},\ q=r^{2}-|c|^{2}}
8594:"Cyclic Averages of Regular Polygons and Platonic Solids"
6733:. A circle is the special case of a supercircle in which
6476:{\displaystyle {\frac {|AP|}{|BP|}}={\frac {|AC|}{|BC|}}}
5056:{\displaystyle {\frac {dy}{dx}}=-{\frac {x_{1}}{y_{1}}}.}
2973:{\displaystyle (x_{1},y_{1}),(x_{2},y_{2}),(x_{3},y_{3})}
1830:
The ratio of a circle's circumference to its diameter is
8208:. History.mcs.st-andrews.ac.uk. Retrieved on 2012-05-03.
8130:. History.mcs.st-andrews.ac.uk. Retrieved on 2012-05-03.
6832:
are taken. The locus of points such that the sum of the
4568:, and the result is that the equation of the tangent is
5716:{\displaystyle 2\angle {CAB}=\angle {DOE}-\angle {BOC}}
5561:
If the angle subtended by the chord at the centre is 90
5126:
enclosed and the square of its radius are proportional.
4203:{\displaystyle pz{\overline {z}}+gz+{\overline {gz}}=q}
2850:
of the line passing through the centre parallel to the
2870:
The equation of the circle determined by three points
1578:), meaning "hoop" or "ring". The origins of the words
1384:: the point equidistant from all points on the circle.
7595:
7543:
7523:
7480:
7334:
7177:
7031:
6908:
6888:
6861:
6838:
6811:
6791:
6770:
6639:
6510:
Inscription in or circumscription about other figures
6389:
6293:
6157:
6109:, a line.) That circle is sometimes said to be drawn
5847:{\displaystyle r={\frac {y^{2}}{8x}}+{\frac {x}{2}}.}
5799:
5661:
5629:
5597:
4999:
4937:
4842:
4717:
4574:
4330:
4234:
4150:
4100:
4053:
3913:
3854:
3767:
3735:
3628:
3498:
2992:
2876:
2699:
2567:
2496:
2386:
2255:
2205:
2133:
2007:
1951:
1866:
1450:
1430:
1313:
that are at a given distance from a given point, the
8273:. Rizzoli International Publications, Incorporated.
8162:: A History of Man's Changing Vision of the Universe
5727:
is the centre of the circle (secant–secant theorem).
4814:{\displaystyle (x_{1}-a)(x-a)+(y_{1}-b)(y-b)=r^{2}.}
4228:. This becomes the above equation for a circle with
3903:
In the general case, the equation can be solved for
1444:
and required to be a positive number. A circle with
6548:that is tangent to each side of the polygon. Every
1755:cannot be performed with straightedge and compass.
130:
118:
104:
94:
41:
8693:
8660:
8422:"Introduction to the theory of analytic functions"
8295:
8189:Tr. Thomas Taylor (1816) Vol. 2, Ch. 2, "Of Plato"
7708:metrics does not generalize to higher dimensions.
7665:A circle of radius 1 (using this distance) is the
7654:
7581:
7529:
7493:
7452:
7316:
7044:
7017:
6894:
6874:
6847:
6824:
6797:
6776:
6703:
6475:
6343:
6228:extended bisects the corresponding exterior angle
6207:{\displaystyle {\frac {AP}{BP}}={\frac {AC}{BC}}.}
6206:
5846:
5715:
5647:
5615:
5055:
4985:
4914:
4813:
4703:
4423:
4316:
4202:
4134:
4084:
4019:
3893:
3792:
3753:
3715:
3576:
3469:
2972:
2828:
2646:
2538:
2452:
2290:
2223:
2155:
2068:
1987:
1896:
1803:, a rainbow, mandalas, rose windows and so forth.
1516:, that is, not containing their boundaries, or as
1512:All of the specified regions may be considered as
1462:
1436:
1364:: a ring-shaped object, the region bounded by two
8644:: CS1 maint: DOI inactive as of September 2024 (
6694:
6665:
6498:are given distinct points in the plane, then the
6051:(it will also pass through the other two points).
5244:The diameter is the longest chord of the circle.
2980:not on a line is obtained by a conversion of the
1325:. A circle bounds a region of the plane called a
8302:(2nd ed.). Addison Wesley Longman. p.
7690:) on a plane is also a square with side length 2
7077:Squaring the circle is the problem, proposed by
2858:). However, this parameterisation works only if
2693:An alternative parametrisation of the circle is
1640:. Disc-shaped prehistoric artifacts include the
1522:
8383:, Dover, 2nd edition, 1996: pp. 104–105, #4–23.
6097:(other than 1) of distances to two fixed foci,
3577:{\displaystyle x^{2}+y^{2}-2axz-2byz+cz^{2}=0.}
1689:
8781:"Interactive Standard Form Equation of Circle"
8598:Communications in Mathematics and Applications
6602:approaches infinity. This fact was applied by
5958:Construction through three noncollinear points
2191:Using radians, the formula for the arc length
6330:
6296:
5451:The chord theorem states that if two chords,
3811:is the anticlockwise angle from the positive
2291:{\displaystyle A={\frac {1}{2}}\theta r^{2}.}
1278:
8:
6552:and every triangle is a tangential polygon.
1751:, proving that the millennia-old problem of
1531:Chord, secant, tangent, radius, and diameter
8538:Tangential Polygon – from Wolfram MathWorld
8397:29(4), September 1998, p. 331, problem 635.
8271:The Louvre Abu Dhabi: A World Vision of Art
7517:. Thus, the value of a geometric analog to
6805:points the vertices of the regular polygon
4094:In parametric form, this can be written as
3492:with the equation of a circle has the form
8298:A History of Mathematics / An Introduction
8245:"Why Did Wassily Kandinsky Paint Circles?"
6252:is a right angle forms a circle, of which
2453:{\displaystyle (x-a)^{2}+(y-b)^{2}=r^{2}.}
1988:{\displaystyle \mathrm {Area} =\pi r^{2}.}
1285:
1271:
1000:
519:
154:
143:
8609:
8348:
7602:
7594:
7568:
7560:
7552:
7544:
7542:
7522:
7481:
7479:
7439:
7429:
7405:
7395:
7377:
7367:
7356:
7347:
7333:
7301:
7297:
7286:
7276:
7252:
7242:
7224:
7214:
7190:
7176:
7036:
7030:
6971:
6958:
6939:
6934:
6924:
6913:
6907:
6887:
6866:
6860:
6837:
6816:
6810:
6790:
6769:
6688:
6674:
6659:
6645:
6638:
6465:
6454:
6447:
6436:
6433:
6422:
6411:
6404:
6393:
6390:
6388:
6329:
6328:
6295:
6294:
6292:
6181:
6158:
6156:
5831:
5812:
5806:
5798:
5702:
5685:
5668:
5660:
5630:
5628:
5598:
5596:
5293:and divides the other chord into lengths
5264:and divides the other chord into lengths
5042:
5032:
5026:
5000:
4998:
4974:
4958:
4942:
4936:
4894:
4876:
4869:
4843:
4841:
4802:
4765:
4725:
4716:
4692:
4673:
4657:
4638:
4610:
4582:
4573:
4411:
4395:
4376:
4363:
4351:
4346:
4331:
4329:
4308:
4303:
4294:
4285:
4259:
4233:
4179:
4157:
4149:
4114:
4099:
4068:
4054:
4052:
3988:
3978:
3973:
3960:
3954:
3924:
3912:
3853:
3775:
3766:
3734:
3704:
3691:
3686:
3652:
3633:
3627:
3562:
3516:
3503:
3497:
3452:
3439:
3423:
3410:
3391:
3378:
3362:
3349:
3331:
3318:
3302:
3289:
3270:
3257:
3241:
3228:
3218:
3203:
3189:
3177:
3163:
3148:
3134:
3122:
3108:
3094:
3080:
3068:
3054:
3039:
3025:
3013:
2999:
2993:
2991:
2961:
2948:
2929:
2916:
2897:
2884:
2875:
2810:
2789:
2754:
2736:
2723:
2700:
2698:
2627:
2591:
2568:
2566:
2527:
2514:
2501:
2495:
2441:
2428:
2403:
2385:
2279:
2262:
2254:
2204:
2199:and subtending an angle of measure 𝜃 is
2140:
2132:
2057:
2035:
2025:
2008:
2006:
1976:
1952:
1950:
1865:
1520:, including their respective boundaries.
1449:
1429:
1340:, which, with related inventions such as
7144:Illustrations of unit circles (see also
6020:. (They meet because the points are not
5648:{\displaystyle {\overset {\frown }{BC}}}
5616:{\displaystyle {\overset {\frown }{DE}}}
5232:is equal to the interior opposite angle.
5198:through the centre bisecting a chord is
3894:{\displaystyle r=2a\cos(\theta -\phi ).}
2846:can be interpreted geometrically as the
1786:. Notice also the circular shape of the
1671:(3.16049...) as an approximate value of
8043:
8019:
5098:). The group of rotations alone is the
5086:around the centre for every angle. Its
3606:: 0). These points are called the
2838:In this parameterisation, the ratio of
2156:{\displaystyle \theta ={\frac {s}{r}}.}
1732:, was connected to the divine for most
1239:
1173:
1122:
1051:
1003:
765:
627:
604:
571:
543:
146:
8637:
8060:, Henry George Liddell, Robert Scott,
5909:compass-and-straightedge constructions
5903:Compass and straightedge constructions
5129:The constants of proportionality are 2
5115:A circle circumference and radius are
1632:, and circular elements are common in
385:Straightedge and compass constructions
38:
8519:Circumcircle – from Wolfram MathWorld
7513:. Thus, a circle's circumference is 8
6598:sides has the circle as its limit as
6344:{\displaystyle {\bigl |}{\bigr |}=1.}
5529:and a secant from the external point
7:
8667:. Mineola, N.Y: Dover Publications.
7025:whose centre is the centroid of the
6559:is any convex polygon about which a
6483:is not a circle, but rather a line.
6383:satisfying the Apollonius condition
6124:and a ratio of distances, any point
6070:Apollonius' definition of a circle:
5770:The sagitta is the vertical segment.
4986:{\displaystyle x_{1}x+y_{1}y=r^{2},}
6271:of points in the complex plane. If
5758:(since the central angle is 180°).
3190:
3135:
3026:
3000:
2666:, interpreted geometrically as the
1997:Equivalently, denoting diameter by
8626:from the original on 22 April 2021
7831:Apollonius circle of the excircles
6151:, since the segments are similar:
6033:passing through one of the points
5699:
5682:
5665:
5319:equals the square of the diameter.
4144:The slightly generalised equation
2539:{\displaystyle x^{2}+y^{2}=r^{2}.}
2076:that is, approximately 79% of the
2018:
2015:
2012:
2009:
1962:
1959:
1956:
1953:
1945:multiplied by the radius squared:
27:Simple curve of Euclidean geometry
25:
8500:Incircle – from Wolfram MathWorld
8121:Chronology for 30000 BC to 500 BC
7915:of an orthodiagonal quadrilateral
6027:Construct the circle with centre
5939:Construct the circle with centre
5518:(corollary of the chord theorem).
4836:, then the slope of this line is
3164:
3109:
3081:
3055:
2983:3-point form of a circle equation
351:Noncommutative algebraic geometry
8696:Geometry: a comprehensive course
8381:Challenging Problems in Geometry
6379:, then the collection of points
5922:Construction with given diameter
5580:is the length of the chord, and
5368:intersect at the exterior point
2080:square (whose side is of length
1604:Santa Barbara County, California
47:
8359:10.4169/college.math.j.46.3.162
8337:The College Mathematics Journal
8003:Three points determine a circle
6970:
6617:is a set of points such that a
6375:is the midpoint of the segment
5899:, we find the required result.
5399:is a chord of the circle, then
5256:divides one chord into lengths
2856:Tangent half-angle substitution
2553:The equation can be written in
2111:, and that angle intercepts an
1897:{\displaystyle C=2\pi r=\pi d.}
1807:are part of some traditions of
8622:(inactive 11 September 2024).
8571:10.1080/00029890.2003.11919989
7569:
7561:
7553:
7545:
7343:
7337:
7186:
7180:
6582:The circle can be viewed as a
6578:Limiting case of other figures
6466:
6455:
6448:
6437:
6423:
6412:
6405:
6394:
6325:
6301:
5372:, then denoting the centre as
5254:intersection of any two chords
4792:
4780:
4777:
4758:
4752:
4740:
4737:
4718:
4685:
4666:
4650:
4631:
4622:
4603:
4594:
4575:
4347:
4332:
4304:
4295:
4069:
4055:
4009:
3997:
3948:
3936:
3885:
3873:
3787:
3768:
3748:
3736:
3676:
3664:
3622:, the equation of a circle is
3458:
3432:
3429:
3403:
3397:
3371:
3368:
3342:
3337:
3311:
3308:
3282:
3276:
3250:
3247:
3221:
3209:
3186:
3183:
3160:
3154:
3131:
3128:
3105:
3100:
3077:
3074:
3051:
3045:
3022:
3019:
2996:
2967:
2941:
2935:
2909:
2903:
2877:
2425:
2412:
2400:
2387:
744:- / other-dimensional
1:
8592:Meskhishvili, Mamuka (2020).
8559:American Mathematical Monthly
8243:Lesso, Rosie (15 June 2022).
7589:in Cartesian coordinates and
7093:Lindemann–Weierstrass theorem
4924:This can also be found using
3793:{\displaystyle (r_{0},\phi )}
3729:is the radius of the circle,
2233:and the formula for the area
1474:consisting of a single point.
7494:{\displaystyle {\sqrt {2}}r}
7171:, distance is determined by
6902:is constant is a circle, if
6633:has an equation of the form
6518:a unique circle, called the
6216:Analogously, a line segment
5584:is the radius of the circle.
5391:is tangent to the circle at
4416:
4400:
4381:
4368:
4264:
4189:
4162:
4135:{\displaystyle z=re^{it}+c.}
4039:, a circle with a centre at
2195:of a circular arc of radius
2165:The circular arc is said to
1915:Area enclosed by a circle =
8728:Encyclopedia of Mathematics
8490:, Dover, 2007 (orig. 1952).
8408:Advanced Euclidean Geometry
8394:College Mathematics Journal
7105:algebraic irrational number
6561:circle can be circumscribed
5860:and with sagitta of length
4479:and the circle has centre (
3754:{\displaystyle (r,\theta )}
3608:circular points at infinity
2354:Cartesian coordinate system
2224:{\displaystyle s=\theta r,}
1919:× area of the shaded square
1770:Symbolism and religious use
1602:Circular cave paintings in
8844:
8770:"Interactive Java applets"
8659:Gamelin, Theodore (1999).
8486:Altshiller-Court, Nathan,
8223:Philadelphia Museum of Art
7070:
6586:of various other figures:
6364:
6059:
5789:of a chord and the length
5735:
5443:
5334:is the circle radius, and
5207:If a central angle and an
4564:) determines the value of
4442:
2686:) makes with the positive
2372:is the set of all points (
1922:
1823:
29:
8379:Posamentier and Salkind,
8092:10.1017/S0003598X00049048
7907:Of certain quadrilaterals
7740:upon itself (known as an
7582:{\displaystyle |x|+|y|=1}
7470:= 1. Taxicab circles are
7328:= 2, giving the familiar
7107:; that is, it is not the
6764:Consider a finite set of
5555:(tangent–secant theorem).
5492:, also cut the circle at
2356:, the circle with centre
2336:) = (1.2, −0.5)
1939:area enclosed by a circle
1852:is related to the radius
1848:. Thus the circumference
46:
8663:Introduction to topology
8420:Harkness, James (1898).
8294:Katz, Victor J. (1998).
7988:Line–circle intersection
7789:Circle of antisimilitude
7769:Archimedes' twin circles
7667:von Neumann neighborhood
7095:, which proves that pi (
7086:compass and straightedge
6538:tangential quadrilateral
5073:Isoperimetric inequality
4926:implicit differentiation
4445:Tangent lines to circles
4085:{\displaystyle |z-c|=r.}
3588:complex projective plane
2848:stereographic projection
2328: = 1, centre (
2089:isoperimetric inequality
1624:Prehistoric people made
1541:Arc, sector, and segment
240:Non-Archimedean geometry
8620:10.26713/cma.v11i3.1420
8062:A Greek-English Lexicon
7886:Polar circle (geometry)
7748:Specially named circles
7324:In Euclidean geometry,
6855:-th power of distances
6752:curve of constant width
6546:circle can be inscribed
5746:Inscribed-angle theorem
5738:Inscribed angle theorem
3486:homogeneous coordinates
2559:trigonometric functions
1935:Measurement of a Circle
1741:Ferdinand von Lindemann
1730:astrology and astronomy
346:Noncommutative geometry
85: centre or origin
32:Circle (disambiguation)
8475:Excursions in Geometry
7876:Orthocentroidal circle
7712:Topological definition
7662:in polar coordinates.
7656:
7583:
7531:
7495:
7454:
7318:
7161:
7046:
7019:
6929:
6896:
6876:
6849:
6826:
6799:
6778:
6705:
6477:
6345:
6208:
6138:angle bisector theorem
6087:
6001:perpendicular bisector
5986:perpendicular bisector
5954:
5911:resulting in circles.
5848:
5771:
5747:
5717:
5649:
5617:
5441:
5186:perpendicular bisector
5057:
4987:
4916:
4815:
4705:
4513:), so it has the form
4425:
4318:
4224:is sometimes called a
4204:
4136:
4086:
4021:
3895:
3794:
3755:
3717:
3578:
3471:
2974:
2830:
2648:
2540:
2468:equation of the circle
2454:
2337:
2292:
2225:
2157:
2099:If a circle of radius
2070:
1989:
1920:
1898:
1791:
1710:
1644:and jade discs called
1621:
1606:
1542:
1532:
1464:
1438:
314:Discrete/Combinatorial
8792:"Munching on Circles"
8477:, Dover, 1969, 14–17.
8219:"Circles in a Circle"
7993:List of circle topics
7983:Inversion in a circle
7657:
7584:
7532:
7496:
7455:
7319:
7143:
7101:transcendental number
7047:
7045:{\displaystyle P_{n}}
7020:
6909:
6897:
6877:
6875:{\displaystyle d_{i}}
6850:
6827:
6825:{\displaystyle P_{n}}
6800:
6779:
6760:Locus of constant sum
6706:
6478:
6346:
6209:
6069:
6062:Circles of Apollonius
5952:
5849:
5769:
5745:
5718:
5650:
5618:
5440:Secant–secant theorem
5439:
5058:
4988:
4917:
4816:
4706:
4426:
4319:
4205:
4137:
4087:
4022:
3896:
3795:
3756:
3718:
3579:
3472:
2975:
2831:
2649:
2541:
2455:
2323:
2316:Cartesian coordinates
2293:
2226:
2158:
2071:
1990:
1914:
1899:
1777:
1612:
1601:
1590:are closely related.
1540:
1530:
1465:
1439:
297:Discrete differential
8410:, Dover Publ., 2007.
7978:Gauss circle problem
7593:
7541:
7530:{\displaystyle \pi }
7521:
7478:
7332:
7175:
7057:equilateral triangle
7029:
6906:
6886:
6859:
6836:
6809:
6789:
6768:
6723:. A supercircle has
6637:
6387:
6353:Stated another way,
6291:
6155:
6056:Circle of Apollonius
5797:
5659:
5627:
5595:
5533:meets the circle at
5525:meets the circle at
5226:cyclic quadrilateral
4997:
4935:
4840:
4715:
4572:
4328:
4232:
4148:
4098:
4051:
3911:
3852:
3765:
3733:
3626:
3496:
2990:
2874:
2697:
2565:
2494:
2384:
2341:Equation of a circle
2300:In the special case
2253:
2203:
2131:
2005:
1949:
1864:
1448:
1428:
58: circumference
8692:Pedoe, Dan (1988).
8438:1899Natur..59..386B
8406:Johnson, Roger A.,
8199:Squaring the circle
8008:Translation of axes
7672:A circle of radius
7073:Squaring the circle
7067:Squaring the circle
7055:In the case of the
6947:
6361:Generalised circles
6220:through some point
6091:Apollonius of Perga
5778:(also known as the
5541:respectively, then
5500:respectively, then
5084:rotational symmetry
5080:reflection symmetry
3983:
3696:
2670:that the ray from (
2662:in the range 0 to 2
2660:parametric variable
2561:sine and cosine as
2472:Pythagorean theorem
2470:, follows from the
1809:Western esotericism
1758:With the advent of
1753:squaring the circle
1463:{\displaystyle r=0}
564:Pythagorean theorem
8752:Weisstein, Eric W.
8543:2013-09-03 at the
8524:2012-01-20 at the
8505:2012-01-21 at the
8471:Ogilvy, C. Stanley
8460:on 7 October 2008.
8204:2008-06-24 at the
8185:2017-01-23 at the
8126:2008-03-22 at the
8056:2013-11-06 at the
7943:Villarceau circles
7920:Of a conic section
7913:Eight-point circle
7764:Archimedean circle
7759:Apollonian circles
7716:The circle is the
7678:Chebyshev distance
7652:
7579:
7527:
7491:
7450:
7314:
7162:
7042:
7015:
6930:
6892:
6872:
6848:{\displaystyle 2m}
6845:
6822:
6795:
6774:
6701:
6534:tangential polygon
6473:
6367:Generalised circle
6341:
6204:
6088:
5955:
5844:
5772:
5748:
5713:
5645:
5613:
5442:
5384:are supplementary.
5053:
4983:
4912:
4811:
4701:
4421:
4314:
4226:generalised circle
4200:
4132:
4082:
4017:
3969:
3891:
3825:, this reduces to
3790:
3751:
3713:
3682:
3574:
3467:
3194:
3168:
3139:
3113:
3085:
3059:
3030:
3004:
2970:
2826:
2824:
2644:
2642:
2536:
2450:
2338:
2288:
2221:
2153:
2103:is centred at the
2066:
1985:
1921:
1894:
1792:
1622:
1613:Circles in an old
1607:
1543:
1533:
1460:
1434:
1305:consisting of all
8818:Elementary shapes
8335:is a constant?".
8313:978-0-321-01618-8
7953:
7952:
7896:Van Lamoen circle
7871:Nine-point circle
7650:
7486:
7445:
7103:, rather than an
6978:
6974:
6973: where
6895:{\displaystyle R}
6798:{\displaystyle n}
6777:{\displaystyle n}
6682:
6653:
6565:bicentric polygon
6471:
6428:
6199:
6176:
6140:the line segment
5839:
5826:
5785:Given the length
5643:
5611:
5364:and a tangent at
5159:Riemannian circle
5157:, it becomes the
5048:
5018:
4993:and its slope is
4907:
4861:
4550:. Evaluating at (
4419:
4403:
4384:
4371:
4274:
4267:
4249:
4192:
4165:
4047:has the equation
4012:
3620:polar coordinates
3614:Polar coordinates
3462:
3213:
2817:
2761:
2324:Circle of radius
2270:
2249:of measure 𝜃 is
2173:, which measures
2148:
2113:arc of the circle
2045:
1764:Wassily Kandinsky
1734:medieval scholars
1558:derives from the
1547:
1546:
1437:{\displaystyle r}
1295:
1294:
1260:
1259:
983:List of geometers
666:Three-dimensional
655:
654:
142:
141:
16:(Redirected from
8835:
8799:
8787:
8776:
8765:
8764:
8736:
8711:
8699:
8679:
8678:
8666:
8656:
8650:
8649:
8643:
8635:
8633:
8631:
8613:
8589:
8583:
8582:
8554:
8548:
8535:
8529:
8516:
8510:
8497:
8491:
8488:College Geometry
8484:
8478:
8468:
8462:
8461:
8456:. Archived from
8446:10.1038/059386a0
8417:
8411:
8404:
8398:
8390:
8384:
8377:
8371:
8370:
8352:
8330:
8324:
8318:
8317:
8301:
8291:
8285:
8284:
8266:
8260:
8259:
8257:
8255:
8240:
8234:
8233:
8231:
8229:
8215:
8209:
8196:
8190:
8171:
8165:
8160:The Sleepwalkers
8152:
8146:
8137:
8131:
8118:
8112:
8111:
8086:(336): 430–446.
8071:
8065:
8048:
8031:
8024:
7931:Directrix circle
7861:Malfatti circles
7784:Chromatic circle
7753:
7723:(the 1-sphere).
7661:
7659:
7658:
7653:
7651:
7649:
7648:
7644:
7626:
7622:
7603:
7588:
7586:
7585:
7580:
7572:
7564:
7556:
7548:
7536:
7534:
7533:
7528:
7503:Euclidean metric
7500:
7498:
7497:
7492:
7487:
7482:
7464:taxicab geometry
7459:
7457:
7456:
7451:
7446:
7444:
7443:
7438:
7434:
7433:
7410:
7409:
7404:
7400:
7399:
7382:
7381:
7376:
7372:
7371:
7357:
7352:
7351:
7346:
7323:
7321:
7320:
7315:
7310:
7309:
7305:
7296:
7292:
7291:
7290:
7285:
7281:
7280:
7257:
7256:
7251:
7247:
7246:
7229:
7228:
7223:
7219:
7218:
7195:
7194:
7189:
7159:
7153:
7098:
7061:regular pentagon
7051:
7049:
7048:
7043:
7041:
7040:
7024:
7022:
7021:
7016:
6976:
6975:
6972:
6966:
6965:
6946:
6938:
6928:
6923:
6901:
6899:
6898:
6893:
6881:
6879:
6878:
6873:
6871:
6870:
6854:
6852:
6851:
6846:
6831:
6829:
6828:
6823:
6821:
6820:
6804:
6802:
6801:
6796:
6783:
6781:
6780:
6775:
6739:
6732:
6710:
6708:
6707:
6702:
6693:
6692:
6687:
6683:
6675:
6664:
6663:
6658:
6654:
6646:
6592:regular polygons
6482:
6480:
6479:
6474:
6472:
6470:
6469:
6458:
6452:
6451:
6440:
6434:
6429:
6427:
6426:
6415:
6409:
6408:
6397:
6391:
6350:
6348:
6347:
6342:
6334:
6333:
6300:
6299:
6248:such that angle
6213:
6211:
6210:
6205:
6200:
6198:
6190:
6182:
6177:
6175:
6167:
6159:
6144:will bisect the
6085:
6050:
6044:
6038:
6032:
6019:
6010:
6009:
5995:
5994:
5980:
5974:
5968:
5963:Name the points
5944:
5936:of the diameter.
5935:
5894:
5874:
5853:
5851:
5850:
5845:
5840:
5832:
5827:
5825:
5817:
5816:
5807:
5732:Inscribed angles
5722:
5720:
5719:
5714:
5712:
5695:
5678:
5654:
5652:
5651:
5646:
5644:
5639:
5631:
5622:
5620:
5619:
5614:
5612:
5607:
5599:
5575:
5554:
5517:
5484:If two secants,
5480:
5446:Power of a point
5426:
5420:
5418:
5417:
5414:
5411:
5360:If a tangent at
5318:
5281:
5149:Thought of as a
5136:
5132:
5108:All circles are
5092:orthogonal group
5062:
5060:
5059:
5054:
5049:
5047:
5046:
5037:
5036:
5027:
5019:
5017:
5009:
5001:
4992:
4990:
4989:
4984:
4979:
4978:
4963:
4962:
4947:
4946:
4921:
4919:
4918:
4913:
4908:
4906:
4899:
4898:
4888:
4881:
4880:
4870:
4862:
4860:
4852:
4844:
4835:
4820:
4818:
4817:
4812:
4807:
4806:
4770:
4769:
4730:
4729:
4710:
4708:
4707:
4702:
4697:
4696:
4678:
4677:
4662:
4661:
4643:
4642:
4615:
4614:
4587:
4586:
4549:
4478:
4453:through a point
4430:
4428:
4427:
4422:
4420:
4412:
4404:
4396:
4385:
4377:
4372:
4364:
4356:
4355:
4350:
4335:
4323:
4321:
4320:
4315:
4313:
4312:
4307:
4298:
4290:
4289:
4272:
4268:
4260:
4247:
4209:
4207:
4206:
4201:
4193:
4188:
4180:
4166:
4158:
4141:
4139:
4138:
4133:
4122:
4121:
4091:
4089:
4088:
4083:
4072:
4058:
4026:
4024:
4023:
4018:
4013:
3993:
3992:
3982:
3977:
3965:
3964:
3955:
3929:
3928:
3900:
3898:
3897:
3892:
3847:
3834:
3824:
3799:
3797:
3796:
3791:
3780:
3779:
3760:
3758:
3757:
3752:
3722:
3720:
3719:
3714:
3709:
3708:
3695:
3690:
3657:
3656:
3638:
3637:
3583:
3581:
3580:
3575:
3567:
3566:
3521:
3520:
3508:
3507:
3480:Homogeneous form
3476:
3474:
3473:
3468:
3463:
3461:
3457:
3456:
3444:
3443:
3428:
3427:
3415:
3414:
3396:
3395:
3383:
3382:
3367:
3366:
3354:
3353:
3340:
3336:
3335:
3323:
3322:
3307:
3306:
3294:
3293:
3275:
3274:
3262:
3261:
3246:
3245:
3233:
3232:
3219:
3214:
3212:
3208:
3207:
3195:
3182:
3181:
3169:
3153:
3152:
3140:
3127:
3126:
3114:
3103:
3099:
3098:
3086:
3073:
3072:
3060:
3044:
3043:
3031:
3018:
3017:
3005:
2994:
2979:
2977:
2976:
2971:
2966:
2965:
2953:
2952:
2934:
2933:
2921:
2920:
2902:
2901:
2889:
2888:
2854: axis (see
2835:
2833:
2832:
2827:
2825:
2818:
2816:
2815:
2814:
2798:
2790:
2762:
2760:
2759:
2758:
2742:
2741:
2740:
2724:
2665:
2653:
2651:
2650:
2645:
2643:
2545:
2543:
2542:
2537:
2532:
2531:
2519:
2518:
2506:
2505:
2459:
2457:
2456:
2451:
2446:
2445:
2433:
2432:
2408:
2407:
2306:
2305:
2297:
2295:
2294:
2289:
2284:
2283:
2271:
2263:
2244:
2236:
2230:
2228:
2227:
2222:
2198:
2194:
2179:
2178:
2162:
2160:
2159:
2154:
2149:
2141:
2122:
2102:
2075:
2073:
2072:
2067:
2062:
2061:
2046:
2041:
2040:
2039:
2026:
2021:
1994:
1992:
1991:
1986:
1981:
1980:
1965:
1944:
1925:Area of a circle
1918:
1903:
1901:
1900:
1895:
1847:
1846:
1833:
1815:Analytic results
1746:
1708:
1675:
1670:
1668:
1667:
1664:
1661:
1523:
1469:
1467:
1466:
1461:
1443:
1441:
1440:
1435:
1287:
1280:
1273:
1001:
520:
453:Zero-dimensional
158:
144:
138:
126:
113:
84:
75:
66:
57:
51:
39:
21:
8843:
8842:
8838:
8837:
8836:
8834:
8833:
8832:
8803:
8802:
8790:
8779:
8768:
8750:
8749:
8721:
8718:
8708:
8691:
8688:
8686:Further reading
8683:
8682:
8675:
8658:
8657:
8653:
8636:
8629:
8627:
8591:
8590:
8586:
8556:
8555:
8551:
8545:Wayback Machine
8536:
8532:
8526:Wayback Machine
8517:
8513:
8507:Wayback Machine
8498:
8494:
8485:
8481:
8469:
8465:
8419:
8418:
8414:
8405:
8401:
8391:
8387:
8378:
8374:
8334:
8328:
8326:
8325:
8321:
8314:
8293:
8292:
8288:
8281:
8268:
8267:
8263:
8253:
8251:
8242:
8241:
8237:
8227:
8225:
8217:
8216:
8212:
8206:Wayback Machine
8197:
8193:
8187:Wayback Machine
8172:
8168:
8155:Arthur Koestler
8153:
8149:
8138:
8134:
8128:Wayback Machine
8119:
8115:
8073:
8072:
8068:
8058:Wayback Machine
8049:
8045:
8040:
8035:
8034:
8025:
8021:
8016:
7959:
7954:
7939:
7926:Director circle
7922:
7909:
7901:
7827:
7819:
7804:Johnson circles
7799:Geodesic circle
7750:
7742:ambient isotopy
7718:one-dimensional
7714:
7707:
7700:
7687:
7669:of its centre.
7634:
7630:
7612:
7608:
7607:
7591:
7590:
7539:
7538:
7519:
7518:
7476:
7475:
7425:
7421:
7420:
7391:
7387:
7386:
7363:
7359:
7358:
7336:
7335:
7330:
7329:
7272:
7268:
7267:
7238:
7234:
7233:
7210:
7206:
7205:
7204:
7200:
7199:
7179:
7178:
7173:
7172:
7155:
7149:
7148:) in different
7138:
7129:
7127:Generalizations
7096:
7075:
7069:
7032:
7027:
7026:
6954:
6904:
6903:
6884:
6883:
6862:
6857:
6856:
6834:
6833:
6812:
6807:
6806:
6787:
6786:
6766:
6765:
6762:
6734:
6724:
6670:
6669:
6641:
6640:
6635:
6634:
6580:
6550:regular polygon
6544:within which a
6512:
6453:
6435:
6410:
6392:
6385:
6384:
6369:
6363:
6289:
6288:
6265:
6256:is a diameter.
6191:
6183:
6168:
6160:
6153:
6152:
6084:
6077:
6071:
6064:
6058:
6046:
6040:
6034:
6028:
6015:
6005:
6004:
6003:of the segment
5990:
5989:
5988:of the segment
5976:
5970:
5964:
5960:
5940:
5931:
5924:
5907:There are many
5905:
5876:
5865:
5818:
5808:
5795:
5794:
5764:
5740:
5734:
5657:
5656:
5632:
5625:
5624:
5600:
5593:
5592:
5566:
5542:
5501:
5464:
5459:, intersect at
5448:
5434:
5415:
5412:
5409:
5408:
5406:
5400:
5348:
5302:
5273:
5239:Thales' theorem
5209:inscribed angle
5178:
5134:
5130:
5068:
5038:
5028:
5010:
5002:
4995:
4994:
4970:
4954:
4938:
4933:
4932:
4890:
4889:
4872:
4871:
4853:
4845:
4838:
4837:
4830:
4824:
4798:
4761:
4721:
4713:
4712:
4688:
4669:
4653:
4634:
4606:
4578:
4570:
4569:
4563:
4556:
4536:
4521:
4514:
4512:
4505:
4476:
4469:
4462:
4447:
4441:
4345:
4326:
4325:
4302:
4281:
4230:
4229:
4181:
4146:
4145:
4110:
4096:
4095:
4049:
4048:
4033:
3984:
3956:
3920:
3909:
3908:
3850:
3849:
3842:
3836:
3826:
3822:
3816:
3806:
3771:
3763:
3762:
3731:
3730:
3700:
3648:
3629:
3624:
3623:
3616:
3558:
3512:
3499:
3494:
3493:
3482:
3448:
3435:
3419:
3406:
3387:
3374:
3358:
3345:
3341:
3327:
3314:
3298:
3285:
3266:
3253:
3237:
3224:
3220:
3199:
3173:
3144:
3118:
3104:
3090:
3064:
3035:
3009:
2995:
2988:
2987:
2957:
2944:
2925:
2912:
2893:
2880:
2872:
2871:
2868:
2823:
2822:
2806:
2799:
2791:
2773:
2767:
2766:
2750:
2743:
2732:
2725:
2707:
2695:
2694:
2663:
2641:
2640:
2611:
2605:
2604:
2575:
2563:
2562:
2555:parametric form
2551:
2549:Parametric form
2523:
2510:
2497:
2492:
2491:
2466:, known as the
2437:
2424:
2399:
2382:
2381:
2343:
2318:
2313:
2303:
2301:
2275:
2251:
2250:
2242:
2239:circular sector
2234:
2201:
2200:
2196:
2192:
2176:
2174:
2129:
2128:
2120:
2100:
2097:
2053:
2031:
2027:
2003:
2002:
1972:
1947:
1946:
1942:
1927:
1916:
1909:
1862:
1861:
1844:
1842:
1831:
1828:
1822:
1817:
1772:
1744:
1724:, particularly
1709:
1696:
1673:
1665:
1662:
1659:
1658:
1656:
1596:
1562:κίρκος/κύκλος (
1552:
1472:degenerate case
1446:
1445:
1426:
1425:
1358:
1291:
1262:
1261:
998:
997:
988:
987:
778:
777:
761:
760:
746:
745:
733:
732:
669:
668:
657:
656:
517:
516:
514:Two-dimensional
505:
504:
478:
477:
475:One-dimensional
466:
465:
456:
455:
444:
443:
377:
376:
375:
358:
357:
206:
205:
194:
171:
136:
124:
111:
90:
89:
82:
80:
73:
71:
67: diameter
64:
62:
55:
35:
28:
23:
22:
15:
12:
11:
5:
8841:
8839:
8831:
8830:
8825:
8823:Conic sections
8820:
8815:
8805:
8804:
8801:
8800:
8788:
8777:
8766:
8747:
8737:
8717:
8716:External links
8714:
8713:
8712:
8706:
8687:
8684:
8681:
8680:
8673:
8651:
8584:
8565:(6): 516–526.
8549:
8530:
8511:
8492:
8479:
8463:
8412:
8399:
8385:
8372:
8343:(3): 162–171.
8332:
8319:
8312:
8286:
8279:
8261:
8235:
8210:
8191:
8166:
8147:
8132:
8113:
8066:
8042:
8041:
8039:
8036:
8033:
8032:
8026:Also known as
8018:
8017:
8015:
8012:
8011:
8010:
8005:
8000:
7995:
7990:
7985:
7980:
7975:
7973:Circle fitting
7970:
7965:
7958:
7955:
7951:
7950:
7946:
7945:
7938:
7935:
7934:
7933:
7928:
7921:
7918:
7917:
7916:
7908:
7905:
7903:
7899:
7898:
7893:
7891:Spieker circle
7888:
7883:
7878:
7873:
7868:
7866:Mandart circle
7863:
7858:
7853:
7851:Lemoine circle
7848:
7843:
7838:
7836:Brocard circle
7833:
7826:
7823:
7821:
7817:
7816:
7811:
7809:Schoch circles
7806:
7801:
7796:
7791:
7786:
7781:
7779:Carlyle circle
7776:
7774:Bankoff circle
7771:
7766:
7761:
7751:
7749:
7746:
7732:homeomorphisms
7713:
7710:
7705:
7698:
7685:
7647:
7643:
7640:
7637:
7633:
7629:
7625:
7621:
7618:
7615:
7611:
7606:
7601:
7598:
7578:
7575:
7571:
7567:
7563:
7559:
7555:
7551:
7547:
7526:
7490:
7485:
7449:
7442:
7437:
7432:
7428:
7424:
7419:
7416:
7413:
7408:
7403:
7398:
7394:
7390:
7385:
7380:
7375:
7370:
7366:
7362:
7355:
7350:
7345:
7342:
7339:
7313:
7308:
7304:
7300:
7295:
7289:
7284:
7279:
7275:
7271:
7266:
7263:
7260:
7255:
7250:
7245:
7241:
7237:
7232:
7227:
7222:
7217:
7213:
7209:
7203:
7198:
7193:
7188:
7185:
7182:
7137:
7130:
7128:
7125:
7071:Main article:
7068:
7065:
7039:
7035:
7014:
7011:
7008:
7005:
7002:
6999:
6996:
6993:
6990:
6987:
6984:
6981:
6969:
6964:
6961:
6957:
6953:
6950:
6945:
6942:
6937:
6933:
6927:
6922:
6919:
6916:
6912:
6891:
6869:
6865:
6844:
6841:
6819:
6815:
6794:
6773:
6761:
6758:
6757:
6756:
6748:
6741:
6700:
6697:
6691:
6686:
6681:
6678:
6673:
6668:
6662:
6657:
6652:
6649:
6644:
6627:
6615:Cartesian oval
6611:
6590:The series of
6579:
6576:
6557:cyclic polygon
6542:convex polygon
6511:
6508:
6468:
6464:
6461:
6457:
6450:
6446:
6443:
6439:
6432:
6425:
6421:
6418:
6414:
6407:
6403:
6400:
6396:
6362:
6359:
6340:
6337:
6332:
6327:
6324:
6321:
6318:
6315:
6312:
6309:
6306:
6303:
6298:
6264:
6261:
6203:
6197:
6194:
6189:
6186:
6180:
6174:
6171:
6166:
6163:
6146:interior angle
6082:
6075:
6057:
6054:
6053:
6052:
6025:
6012:
5999:Construct the
5997:
5984:Construct the
5982:
5959:
5956:
5947:
5946:
5937:
5927:Construct the
5923:
5920:
5904:
5901:
5895:. Solving for
5843:
5838:
5835:
5830:
5824:
5821:
5815:
5811:
5805:
5802:
5763:
5760:
5733:
5730:
5729:
5728:
5711:
5708:
5705:
5701:
5698:
5694:
5691:
5688:
5684:
5681:
5677:
5674:
5671:
5667:
5664:
5642:
5638:
5635:
5610:
5606:
5603:
5585:
5559:
5556:
5519:
5482:
5433:
5430:
5429:
5428:
5385:
5376:, the angles ∠
5358:
5355:
5352:
5347:
5344:
5343:
5342:
5339:
5320:
5283:
5250:
5249:
5248:
5242:
5235:
5234:
5233:
5230:exterior angle
5215:
5212:
5205:
5204:
5203:
5192:
5182:
5177:
5174:
5173:
5172:
5164:
5163:
5162:
5140:
5139:
5138:
5127:
5120:
5106:
5088:symmetry group
5076:
5067:
5064:
5052:
5045:
5041:
5035:
5031:
5025:
5022:
5016:
5013:
5008:
5005:
4982:
4977:
4973:
4969:
4966:
4961:
4957:
4953:
4950:
4945:
4941:
4911:
4905:
4902:
4897:
4893:
4887:
4884:
4879:
4875:
4868:
4865:
4859:
4856:
4851:
4848:
4828:
4810:
4805:
4801:
4797:
4794:
4791:
4788:
4785:
4782:
4779:
4776:
4773:
4768:
4764:
4760:
4757:
4754:
4751:
4748:
4745:
4742:
4739:
4736:
4733:
4728:
4724:
4720:
4700:
4695:
4691:
4687:
4684:
4681:
4676:
4672:
4668:
4665:
4660:
4656:
4652:
4649:
4646:
4641:
4637:
4633:
4630:
4627:
4624:
4621:
4618:
4613:
4609:
4605:
4602:
4599:
4596:
4593:
4590:
4585:
4581:
4577:
4561:
4554:
4534:
4519:
4510:
4503:
4474:
4467:
4443:Main article:
4440:
4437:
4418:
4415:
4410:
4407:
4402:
4399:
4394:
4391:
4388:
4383:
4380:
4375:
4370:
4367:
4362:
4359:
4354:
4349:
4344:
4341:
4338:
4334:
4311:
4306:
4301:
4297:
4293:
4288:
4284:
4280:
4277:
4271:
4266:
4263:
4258:
4255:
4252:
4246:
4243:
4240:
4237:
4199:
4196:
4191:
4187:
4184:
4178:
4175:
4172:
4169:
4164:
4161:
4156:
4153:
4131:
4128:
4125:
4120:
4117:
4113:
4109:
4106:
4103:
4081:
4078:
4075:
4071:
4067:
4064:
4061:
4057:
4032:
4029:
4016:
4011:
4008:
4005:
4002:
3999:
3996:
3991:
3987:
3981:
3976:
3972:
3968:
3963:
3959:
3953:
3950:
3947:
3944:
3941:
3938:
3935:
3932:
3927:
3923:
3919:
3916:
3890:
3887:
3884:
3881:
3878:
3875:
3872:
3869:
3866:
3863:
3860:
3857:
3840:
3820:
3804:
3789:
3786:
3783:
3778:
3774:
3770:
3750:
3747:
3744:
3741:
3738:
3712:
3707:
3703:
3699:
3694:
3689:
3685:
3681:
3678:
3675:
3672:
3669:
3666:
3663:
3660:
3655:
3651:
3647:
3644:
3641:
3636:
3632:
3615:
3612:
3573:
3570:
3565:
3561:
3557:
3554:
3551:
3548:
3545:
3542:
3539:
3536:
3533:
3530:
3527:
3524:
3519:
3515:
3511:
3506:
3502:
3481:
3478:
3466:
3460:
3455:
3451:
3447:
3442:
3438:
3434:
3431:
3426:
3422:
3418:
3413:
3409:
3405:
3402:
3399:
3394:
3390:
3386:
3381:
3377:
3373:
3370:
3365:
3361:
3357:
3352:
3348:
3344:
3339:
3334:
3330:
3326:
3321:
3317:
3313:
3310:
3305:
3301:
3297:
3292:
3288:
3284:
3281:
3278:
3273:
3269:
3265:
3260:
3256:
3252:
3249:
3244:
3240:
3236:
3231:
3227:
3223:
3217:
3211:
3206:
3202:
3198:
3193:
3188:
3185:
3180:
3176:
3172:
3167:
3162:
3159:
3156:
3151:
3147:
3143:
3138:
3133:
3130:
3125:
3121:
3117:
3112:
3107:
3102:
3097:
3093:
3089:
3084:
3079:
3076:
3071:
3067:
3063:
3058:
3053:
3050:
3047:
3042:
3038:
3034:
3029:
3024:
3021:
3016:
3012:
3008:
3003:
2998:
2969:
2964:
2960:
2956:
2951:
2947:
2943:
2940:
2937:
2932:
2928:
2924:
2919:
2915:
2911:
2908:
2905:
2900:
2896:
2892:
2887:
2883:
2879:
2867:
2864:
2821:
2813:
2809:
2805:
2802:
2797:
2794:
2788:
2785:
2782:
2779:
2776:
2774:
2772:
2769:
2768:
2765:
2757:
2753:
2749:
2746:
2739:
2735:
2731:
2728:
2722:
2719:
2716:
2713:
2710:
2708:
2706:
2703:
2702:
2639:
2636:
2633:
2630:
2626:
2623:
2620:
2617:
2614:
2612:
2610:
2607:
2606:
2603:
2600:
2597:
2594:
2590:
2587:
2584:
2581:
2578:
2576:
2574:
2571:
2570:
2550:
2547:
2535:
2530:
2526:
2522:
2517:
2513:
2509:
2504:
2500:
2449:
2444:
2440:
2436:
2431:
2427:
2423:
2420:
2417:
2414:
2411:
2406:
2402:
2398:
2395:
2392:
2389:
2342:
2339:
2317:
2314:
2312:
2309:
2287:
2282:
2278:
2274:
2269:
2266:
2261:
2258:
2220:
2217:
2214:
2211:
2208:
2171:complete angle
2152:
2147:
2144:
2139:
2136:
2096:
2093:
2078:circumscribing
2065:
2060:
2056:
2052:
2049:
2044:
2038:
2034:
2030:
2024:
2020:
2017:
2014:
2011:
1984:
1979:
1975:
1971:
1968:
1964:
1961:
1958:
1955:
1923:Main article:
1908:
1905:
1893:
1890:
1887:
1884:
1881:
1878:
1875:
1872:
1869:
1824:Main article:
1821:
1818:
1816:
1813:
1771:
1768:
1749:transcendental
1718:Seventh Letter
1694:
1642:Nebra sky disc
1638:cave paintings
1630:timber circles
1595:
1592:
1551:
1548:
1545:
1544:
1534:
1510:
1509:
1503:
1497:
1491:
1481:
1475:
1459:
1456:
1453:
1433:
1419:
1413:
1407:
1401:
1391:
1385:
1379:
1369:
1357:
1354:
1293:
1292:
1290:
1289:
1282:
1275:
1267:
1264:
1263:
1258:
1257:
1256:
1255:
1250:
1242:
1241:
1237:
1236:
1235:
1234:
1229:
1224:
1219:
1214:
1209:
1204:
1199:
1194:
1189:
1184:
1176:
1175:
1171:
1170:
1169:
1168:
1163:
1158:
1153:
1148:
1143:
1138:
1133:
1125:
1124:
1120:
1119:
1118:
1117:
1112:
1107:
1102:
1097:
1092:
1087:
1082:
1077:
1072:
1067:
1062:
1054:
1053:
1049:
1048:
1047:
1046:
1041:
1036:
1031:
1026:
1021:
1016:
1008:
1007:
999:
995:
994:
993:
990:
989:
986:
985:
980:
975:
970:
965:
960:
955:
950:
945:
940:
935:
930:
925:
920:
915:
910:
905:
900:
895:
890:
885:
880:
875:
870:
865:
860:
855:
850:
845:
840:
835:
830:
825:
820:
815:
810:
805:
800:
795:
790:
785:
779:
775:
774:
773:
770:
769:
763:
762:
759:
758:
753:
747:
740:
739:
738:
735:
734:
731:
730:
725:
720:
718:Platonic Solid
715:
710:
705:
700:
695:
690:
689:
688:
677:
676:
670:
664:
663:
662:
659:
658:
653:
652:
651:
650:
645:
640:
632:
631:
625:
624:
623:
622:
617:
609:
608:
602:
601:
600:
599:
594:
589:
584:
576:
575:
569:
568:
567:
566:
561:
556:
548:
547:
541:
540:
539:
538:
533:
528:
518:
512:
511:
510:
507:
506:
503:
502:
497:
496:
495:
490:
479:
473:
472:
471:
468:
467:
464:
463:
457:
451:
450:
449:
446:
445:
442:
441:
436:
431:
425:
424:
419:
414:
404:
399:
394:
388:
387:
378:
374:
373:
370:
366:
365:
364:
363:
360:
359:
356:
355:
354:
353:
343:
338:
333:
328:
323:
322:
321:
311:
306:
301:
300:
299:
294:
289:
279:
278:
277:
272:
262:
257:
252:
247:
242:
237:
236:
235:
230:
229:
228:
213:
207:
201:
200:
199:
196:
195:
193:
192:
182:
176:
173:
172:
159:
151:
150:
140:
139:
134:
128:
127:
122:
116:
115:
108:
106:Symmetry group
102:
101:
96:
92:
91:
81:
72:
63:
54:
52:
44:
43:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
8840:
8829:
8826:
8824:
8821:
8819:
8816:
8814:
8811:
8810:
8808:
8797:
8793:
8789:
8786:
8782:
8778:
8775:
8771:
8767:
8762:
8761:
8756:
8753:
8748:
8745:
8741:
8738:
8734:
8730:
8729:
8724:
8720:
8719:
8715:
8709:
8707:9780486658124
8703:
8698:
8697:
8690:
8689:
8685:
8676:
8670:
8665:
8664:
8655:
8652:
8647:
8641:
8625:
8621:
8617:
8612:
8607:
8603:
8599:
8595:
8588:
8585:
8580:
8576:
8572:
8568:
8564:
8560:
8553:
8550:
8546:
8542:
8539:
8534:
8531:
8527:
8523:
8520:
8515:
8512:
8508:
8504:
8501:
8496:
8493:
8489:
8483:
8480:
8476:
8472:
8467:
8464:
8459:
8455:
8451:
8447:
8443:
8439:
8435:
8431:
8427:
8423:
8416:
8413:
8409:
8403:
8400:
8396:
8395:
8389:
8386:
8382:
8376:
8373:
8368:
8364:
8360:
8356:
8351:
8346:
8342:
8338:
8323:
8320:
8315:
8309:
8305:
8300:
8299:
8290:
8287:
8282:
8280:9782370741004
8276:
8272:
8265:
8262:
8250:
8246:
8239:
8236:
8224:
8220:
8214:
8211:
8207:
8203:
8200:
8195:
8192:
8188:
8184:
8181:
8180:
8175:
8170:
8167:
8163:
8161:
8156:
8151:
8148:
8145:
8141:
8136:
8133:
8129:
8125:
8122:
8117:
8114:
8109:
8105:
8101:
8097:
8093:
8089:
8085:
8081:
8077:
8070:
8067:
8063:
8059:
8055:
8052:
8047:
8044:
8037:
8029:
8023:
8020:
8013:
8009:
8006:
8004:
8001:
7999:
7996:
7994:
7991:
7989:
7986:
7984:
7981:
7979:
7976:
7974:
7971:
7969:
7966:
7964:
7963:Affine sphere
7961:
7960:
7956:
7949:
7944:
7941:
7940:
7936:
7932:
7929:
7927:
7924:
7923:
7919:
7914:
7911:
7910:
7906:
7904:
7902:
7897:
7894:
7892:
7889:
7887:
7884:
7882:
7879:
7877:
7874:
7872:
7869:
7867:
7864:
7862:
7859:
7857:
7856:Lester circle
7854:
7852:
7849:
7847:
7844:
7842:
7839:
7837:
7834:
7832:
7829:
7828:
7825:Of a triangle
7824:
7822:
7820:
7815:
7812:
7810:
7807:
7805:
7802:
7800:
7797:
7795:
7792:
7790:
7787:
7785:
7782:
7780:
7777:
7775:
7772:
7770:
7767:
7765:
7762:
7760:
7757:
7756:
7755:
7754:
7747:
7745:
7743:
7739:
7738:
7733:
7729:
7724:
7722:
7719:
7711:
7709:
7704:
7697:
7693:
7689:
7684:
7679:
7675:
7670:
7668:
7663:
7645:
7641:
7638:
7635:
7631:
7627:
7623:
7619:
7616:
7613:
7609:
7604:
7599:
7596:
7576:
7573:
7565:
7557:
7549:
7524:
7516:
7512:
7508:
7504:
7488:
7483:
7473:
7469:
7465:
7460:
7447:
7440:
7435:
7430:
7426:
7422:
7417:
7414:
7411:
7406:
7401:
7396:
7392:
7388:
7383:
7378:
7373:
7368:
7364:
7360:
7353:
7348:
7340:
7327:
7311:
7306:
7302:
7298:
7293:
7287:
7282:
7277:
7273:
7269:
7264:
7261:
7258:
7253:
7248:
7243:
7239:
7235:
7230:
7225:
7220:
7215:
7211:
7207:
7201:
7196:
7191:
7183:
7170:
7168:
7158:
7152:
7147:
7142:
7135:
7131:
7126:
7124:
7123:enthusiasts.
7122:
7118:
7114:
7110:
7106:
7102:
7094:
7089:
7087:
7083:
7080:
7074:
7066:
7064:
7062:
7058:
7053:
7037:
7033:
7012:
7009:
7006:
7003:
7000:
6997:
6994:
6991:
6988:
6985:
6982:
6979:
6967:
6962:
6959:
6955:
6951:
6948:
6943:
6940:
6935:
6931:
6925:
6920:
6917:
6914:
6910:
6889:
6867:
6863:
6842:
6839:
6817:
6813:
6792:
6771:
6759:
6753:
6749:
6746:
6742:
6737:
6731:
6727:
6722:
6718:
6714:
6711:for positive
6698:
6695:
6689:
6684:
6679:
6676:
6671:
6666:
6660:
6655:
6650:
6647:
6642:
6632:
6628:
6624:
6620:
6616:
6612:
6609:
6608:approximate π
6605:
6601:
6597:
6593:
6589:
6588:
6587:
6585:
6584:limiting case
6577:
6575:
6573:
6568:
6566:
6562:
6558:
6553:
6551:
6547:
6543:
6539:
6535:
6530:
6528:
6523:
6521:
6517:
6509:
6507:
6505:
6501:
6497:
6493:
6489:
6484:
6462:
6459:
6444:
6441:
6430:
6419:
6416:
6401:
6398:
6382:
6378:
6374:
6368:
6360:
6358:
6356:
6351:
6338:
6335:
6322:
6319:
6316:
6313:
6310:
6307:
6304:
6286:
6282:
6278:
6274:
6270:
6262:
6260:
6257:
6255:
6251:
6247:
6243:
6239:
6235:
6231:
6227:
6223:
6219:
6214:
6201:
6195:
6192:
6187:
6184:
6178:
6172:
6169:
6164:
6161:
6150:
6147:
6143:
6139:
6135:
6131:
6127:
6123:
6119:
6114:
6112:
6108:
6104:
6100:
6096:
6092:
6081:
6074:
6068:
6063:
6055:
6049:
6043:
6037:
6031:
6026:
6023:
6018:
6013:
6008:
6002:
5998:
5993:
5987:
5983:
5979:
5973:
5967:
5962:
5961:
5957:
5951:
5943:
5938:
5934:
5930:
5926:
5925:
5921:
5919:
5917:
5912:
5910:
5902:
5900:
5898:
5892:
5888:
5884:
5880:
5873:
5869:
5863:
5859:
5854:
5841:
5836:
5833:
5828:
5822:
5819:
5813:
5809:
5803:
5800:
5792:
5788:
5783:
5781:
5777:
5768:
5761:
5759:
5757:
5753:
5752:central angle
5744:
5739:
5731:
5726:
5709:
5706:
5703:
5696:
5692:
5689:
5686:
5679:
5675:
5672:
5669:
5662:
5640:
5636:
5633:
5608:
5604:
5601:
5590:
5586:
5583:
5579:
5573:
5569:
5564:
5560:
5557:
5553:
5549:
5545:
5540:
5536:
5532:
5528:
5524:
5520:
5516:
5512:
5508:
5504:
5499:
5495:
5491:
5487:
5483:
5479:
5475:
5471:
5467:
5462:
5458:
5454:
5450:
5449:
5447:
5438:
5431:
5424:
5404:
5398:
5394:
5390:
5386:
5383:
5379:
5375:
5371:
5367:
5363:
5359:
5356:
5353:
5350:
5349:
5345:
5340:
5337:
5333:
5329:
5325:
5321:
5317:
5313:
5309:
5305:
5300:
5296:
5292:
5288:
5284:
5280:
5276:
5271:
5267:
5263:
5259:
5255:
5251:
5246:
5245:
5243:
5240:
5236:
5231:
5227:
5223:
5222:
5220:
5219:supplementary
5216:
5213:
5210:
5206:
5202:to the chord.
5201:
5200:perpendicular
5197:
5193:
5190:
5189:
5187:
5183:
5180:
5179:
5175:
5170:
5165:
5160:
5156:
5152:
5148:
5147:
5145:
5141:
5137:respectively.
5128:
5125:
5121:
5118:
5114:
5113:
5111:
5107:
5104:
5101:
5097:
5093:
5089:
5085:
5082:, and it has
5081:
5077:
5074:
5070:
5069:
5065:
5063:
5050:
5043:
5039:
5033:
5029:
5023:
5020:
5014:
5011:
5006:
5003:
4980:
4975:
4971:
4967:
4964:
4959:
4955:
4951:
4948:
4943:
4939:
4929:
4927:
4922:
4909:
4903:
4900:
4895:
4891:
4885:
4882:
4877:
4873:
4866:
4863:
4857:
4854:
4849:
4846:
4834:
4827:
4821:
4808:
4803:
4799:
4795:
4789:
4786:
4783:
4774:
4771:
4766:
4762:
4755:
4749:
4746:
4743:
4734:
4731:
4726:
4722:
4698:
4693:
4689:
4682:
4679:
4674:
4670:
4663:
4658:
4654:
4647:
4644:
4639:
4635:
4628:
4625:
4619:
4616:
4611:
4607:
4600:
4597:
4591:
4588:
4583:
4579:
4567:
4560:
4553:
4548:
4544:
4540:
4533:
4529:
4525:
4518:
4509:
4502:
4498:
4494:
4490:
4487:) and radius
4486:
4482:
4473:
4466:
4460:
4456:
4452:
4446:
4439:Tangent lines
4438:
4436:
4434:
4413:
4408:
4405:
4397:
4392:
4389:
4386:
4378:
4373:
4365:
4360:
4357:
4352:
4342:
4339:
4336:
4309:
4299:
4291:
4286:
4282:
4278:
4275:
4269:
4261:
4256:
4253:
4250:
4244:
4241:
4238:
4235:
4227:
4223:
4219:
4215:
4210:
4197:
4194:
4185:
4182:
4176:
4173:
4170:
4167:
4159:
4154:
4151:
4142:
4129:
4126:
4123:
4118:
4115:
4111:
4107:
4104:
4101:
4092:
4079:
4076:
4073:
4065:
4062:
4059:
4046:
4042:
4038:
4037:complex plane
4031:Complex plane
4030:
4028:
4014:
4006:
4003:
4000:
3994:
3989:
3985:
3979:
3974:
3970:
3966:
3961:
3957:
3951:
3945:
3942:
3939:
3933:
3930:
3925:
3921:
3917:
3914:
3906:
3901:
3888:
3882:
3879:
3876:
3870:
3867:
3864:
3861:
3858:
3855:
3846:
3839:
3833:
3829:
3819:
3814:
3810:
3803:
3784:
3781:
3776:
3772:
3745:
3742:
3739:
3728:
3723:
3710:
3705:
3701:
3697:
3692:
3687:
3683:
3679:
3673:
3670:
3667:
3661:
3658:
3653:
3649:
3645:
3642:
3639:
3634:
3630:
3621:
3613:
3611:
3609:
3605:
3601:
3597:
3593:
3590:) the points
3589:
3584:
3571:
3568:
3563:
3559:
3555:
3552:
3549:
3546:
3543:
3540:
3537:
3534:
3531:
3528:
3525:
3522:
3517:
3513:
3509:
3504:
3500:
3491:
3490:conic section
3487:
3479:
3477:
3464:
3453:
3449:
3445:
3440:
3436:
3424:
3420:
3416:
3411:
3407:
3400:
3392:
3388:
3384:
3379:
3375:
3363:
3359:
3355:
3350:
3346:
3332:
3328:
3324:
3319:
3315:
3303:
3299:
3295:
3290:
3286:
3279:
3271:
3267:
3263:
3258:
3254:
3242:
3238:
3234:
3229:
3225:
3215:
3204:
3200:
3196:
3191:
3178:
3174:
3170:
3165:
3157:
3149:
3145:
3141:
3136:
3123:
3119:
3115:
3110:
3095:
3091:
3087:
3082:
3069:
3065:
3061:
3056:
3048:
3040:
3036:
3032:
3027:
3014:
3010:
3006:
3001:
2985:
2984:
2962:
2958:
2954:
2949:
2945:
2938:
2930:
2926:
2922:
2917:
2913:
2906:
2898:
2894:
2890:
2885:
2881:
2865:
2863:
2861:
2857:
2853:
2849:
2845:
2841:
2836:
2819:
2811:
2807:
2803:
2800:
2795:
2792:
2786:
2783:
2780:
2777:
2775:
2770:
2763:
2755:
2751:
2747:
2744:
2737:
2733:
2729:
2726:
2720:
2717:
2714:
2711:
2709:
2704:
2691:
2689:
2685:
2681:
2677:
2673:
2669:
2661:
2657:
2637:
2634:
2631:
2628:
2624:
2621:
2618:
2615:
2613:
2608:
2601:
2598:
2595:
2592:
2588:
2585:
2582:
2579:
2577:
2572:
2560:
2556:
2548:
2546:
2533:
2528:
2524:
2520:
2515:
2511:
2507:
2502:
2498:
2489:
2485:
2481:
2477:
2473:
2469:
2465:
2460:
2447:
2442:
2438:
2434:
2429:
2421:
2418:
2415:
2409:
2404:
2396:
2393:
2390:
2379:
2375:
2371:
2368:) and radius
2367:
2363:
2359:
2355:
2352:
2348:
2340:
2335:
2331:
2327:
2322:
2315:
2310:
2308:
2298:
2285:
2280:
2276:
2272:
2267:
2264:
2259:
2256:
2248:
2247:central angle
2240:
2231:
2218:
2215:
2212:
2209:
2206:
2189:
2187:
2183:
2180:radians, 360
2172:
2168:
2163:
2150:
2145:
2142:
2137:
2134:
2126:
2118:
2114:
2110:
2106:
2094:
2092:
2090:
2085:
2083:
2079:
2063:
2058:
2054:
2050:
2047:
2042:
2036:
2032:
2028:
2022:
2000:
1995:
1982:
1977:
1973:
1969:
1966:
1940:
1936:
1932:
1929:As proved by
1926:
1913:
1907:Area enclosed
1906:
1904:
1891:
1888:
1885:
1882:
1879:
1876:
1873:
1870:
1867:
1859:
1856:and diameter
1855:
1851:
1840:
1837:
1827:
1826:Circumference
1820:Circumference
1819:
1814:
1812:
1810:
1806:
1805:Magic circles
1802:
1796:
1789:
1785:
1781:
1776:
1769:
1767:
1765:
1761:
1756:
1754:
1750:
1742:
1737:
1735:
1731:
1727:
1723:
1719:
1715:
1707:
1703:
1699:
1693:
1688:
1686:
1685:
1678:
1676:
1654:
1653:Rhind papyrus
1651:The Egyptian
1649:
1647:
1643:
1639:
1635:
1631:
1627:
1626:stone circles
1619:
1616:
1611:
1605:
1600:
1593:
1591:
1589:
1588:
1583:
1582:
1577:
1573:
1572:Homeric Greek
1569:
1565:
1564:kirkos/kuklos
1561:
1557:
1549:
1539:
1535:
1529:
1525:
1524:
1521:
1519:
1515:
1507:
1504:
1501:
1498:
1495:
1492:
1489:
1485:
1482:
1479:
1476:
1473:
1457:
1454:
1451:
1431:
1423:
1420:
1417:
1414:
1411:
1408:
1405:
1402:
1399:
1395:
1394:Circumference
1392:
1389:
1386:
1383:
1380:
1377:
1373:
1370:
1367:
1363:
1360:
1359:
1355:
1353:
1351:
1347:
1343:
1339:
1335:
1330:
1328:
1324:
1320:
1316:
1312:
1308:
1304:
1300:
1288:
1283:
1281:
1276:
1274:
1269:
1268:
1266:
1265:
1254:
1251:
1249:
1246:
1245:
1244:
1243:
1238:
1233:
1230:
1228:
1225:
1223:
1220:
1218:
1215:
1213:
1210:
1208:
1205:
1203:
1200:
1198:
1195:
1193:
1190:
1188:
1185:
1183:
1180:
1179:
1178:
1177:
1172:
1167:
1164:
1162:
1159:
1157:
1154:
1152:
1149:
1147:
1144:
1142:
1139:
1137:
1134:
1132:
1129:
1128:
1127:
1126:
1121:
1116:
1113:
1111:
1108:
1106:
1103:
1101:
1098:
1096:
1093:
1091:
1088:
1086:
1083:
1081:
1078:
1076:
1073:
1071:
1068:
1066:
1063:
1061:
1058:
1057:
1056:
1055:
1050:
1045:
1042:
1040:
1037:
1035:
1032:
1030:
1027:
1025:
1022:
1020:
1017:
1015:
1012:
1011:
1010:
1009:
1006:
1002:
992:
991:
984:
981:
979:
976:
974:
971:
969:
966:
964:
961:
959:
956:
954:
951:
949:
946:
944:
941:
939:
936:
934:
931:
929:
926:
924:
921:
919:
916:
914:
911:
909:
906:
904:
901:
899:
896:
894:
891:
889:
886:
884:
881:
879:
876:
874:
871:
869:
866:
864:
861:
859:
856:
854:
851:
849:
846:
844:
841:
839:
836:
834:
831:
829:
826:
824:
821:
819:
816:
814:
811:
809:
806:
804:
801:
799:
796:
794:
791:
789:
786:
784:
781:
780:
772:
771:
768:
764:
757:
754:
752:
749:
748:
743:
737:
736:
729:
726:
724:
721:
719:
716:
714:
711:
709:
706:
704:
701:
699:
696:
694:
691:
687:
684:
683:
682:
679:
678:
675:
672:
671:
667:
661:
660:
649:
646:
644:
643:Circumference
641:
639:
636:
635:
634:
633:
630:
626:
621:
618:
616:
613:
612:
611:
610:
607:
606:Quadrilateral
603:
598:
595:
593:
590:
588:
585:
583:
580:
579:
578:
577:
574:
573:Parallelogram
570:
565:
562:
560:
557:
555:
552:
551:
550:
549:
546:
542:
537:
534:
532:
529:
527:
524:
523:
522:
521:
515:
509:
508:
501:
498:
494:
491:
489:
486:
485:
484:
481:
480:
476:
470:
469:
462:
459:
458:
454:
448:
447:
440:
437:
435:
432:
430:
427:
426:
423:
420:
418:
415:
412:
411:Perpendicular
408:
407:Orthogonality
405:
403:
400:
398:
395:
393:
390:
389:
386:
383:
382:
381:
371:
368:
367:
362:
361:
352:
349:
348:
347:
344:
342:
339:
337:
334:
332:
331:Computational
329:
327:
324:
320:
317:
316:
315:
312:
310:
307:
305:
302:
298:
295:
293:
290:
288:
285:
284:
283:
280:
276:
273:
271:
268:
267:
266:
263:
261:
258:
256:
253:
251:
248:
246:
243:
241:
238:
234:
231:
227:
224:
223:
222:
219:
218:
217:
216:Non-Euclidean
214:
212:
209:
208:
204:
198:
197:
190:
186:
183:
181:
178:
177:
175:
174:
170:
166:
162:
157:
153:
152:
149:
145:
135:
133:
129:
123:
121:
117:
114:
109:
107:
103:
100:
99:Conic section
97:
93:
88:
79:
76: radius
70:
61:
50:
45:
40:
37:
33:
19:
8796:Cut-the-Knot
8784:
8773:
8758:
8726:
8695:
8662:
8654:
8640:cite journal
8628:. Retrieved
8601:
8597:
8587:
8562:
8558:
8552:
8533:
8514:
8495:
8487:
8482:
8474:
8466:
8458:the original
8432:(1530): 30.
8429:
8425:
8415:
8407:
8402:
8392:
8388:
8380:
8375:
8340:
8336:
8322:
8297:
8289:
8270:
8264:
8252:. Retrieved
8249:TheCollector
8248:
8238:
8226:. Retrieved
8222:
8213:
8194:
8178:
8169:
8158:
8150:
8135:
8116:
8083:
8079:
8069:
8064:, on Perseus
8061:
8046:
8022:
7947:
7900:
7881:Parry circle
7818:
7736:
7725:
7715:
7702:
7695:
7691:
7682:
7673:
7671:
7664:
7514:
7510:
7506:
7467:
7461:
7325:
7166:
7163:
7156:
7150:
7146:superellipse
7133:
7090:
7076:
7054:
6763:
6745:Cassini oval
6735:
6729:
6725:
6720:
6716:
6712:
6631:superellipse
6619:weighted sum
6599:
6595:
6581:
6569:
6554:
6536:, such as a
6531:
6524:
6513:
6503:
6495:
6491:
6487:
6485:
6380:
6376:
6372:
6370:
6354:
6352:
6284:
6280:
6276:
6272:
6266:
6263:Cross-ratios
6258:
6253:
6249:
6245:
6241:
6237:
6233:
6229:
6225:
6221:
6217:
6215:
6148:
6141:
6133:
6129:
6125:
6121:
6117:
6115:
6113:two points.
6110:
6106:
6102:
6098:
6094:
6089:
6079:
6072:
6047:
6041:
6035:
6029:
6016:
6006:
5991:
5977:
5971:
5965:
5941:
5932:
5913:
5906:
5896:
5890:
5886:
5882:
5878:
5871:
5867:
5861:
5857:
5855:
5790:
5786:
5784:
5773:
5749:
5724:
5655:). That is,
5588:
5581:
5577:
5571:
5567:
5551:
5547:
5543:
5538:
5534:
5530:
5526:
5522:
5514:
5510:
5506:
5502:
5497:
5493:
5489:
5485:
5477:
5473:
5469:
5465:
5460:
5456:
5452:
5422:
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5396:
5392:
5388:
5381:
5377:
5373:
5369:
5365:
5361:
5335:
5331:
5327:
5323:
5315:
5311:
5307:
5303:
5298:
5294:
5290:
5286:
5278:
5274:
5269:
5265:
5261:
5257:
5196:line segment
5169:circumcircle
5151:great circle
5117:proportional
5102:
5100:circle group
5095:
4930:
4923:
4832:
4825:
4822:
4565:
4558:
4551:
4546:
4542:
4538:
4531:
4527:
4523:
4516:
4507:
4500:
4496:
4492:
4488:
4484:
4480:
4471:
4464:
4458:
4454:
4451:tangent line
4448:
4221:
4220:and complex
4217:
4213:
4211:
4143:
4093:
4044:
4040:
4034:
3904:
3902:
3844:
3837:
3831:
3827:
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3801:
3726:
3724:
3617:
3603:
3599:
3595:
3591:
3585:
3483:
2982:
2869:
2866:3-point form
2859:
2851:
2843:
2839:
2837:
2692:
2690: axis.
2687:
2683:
2679:
2675:
2671:
2655:
2552:
2487:
2483:
2479:
2475:
2467:
2461:
2380:) such that
2377:
2373:
2369:
2365:
2361:
2350:
2346:
2344:
2333:
2329:
2325:
2299:
2232:
2190:
2164:
2098:
2086:
2081:
1998:
1996:
1928:
1857:
1853:
1849:
1829:
1801:Dharma wheel
1797:
1793:
1760:abstract art
1757:
1743:proved that
1739:In 1880 CE,
1738:
1711:
1690:
1683:
1679:
1650:
1623:
1618:astronomical
1585:
1579:
1575:
1566:), itself a
1563:
1555:
1553:
1517:
1513:
1511:
1487:
1331:
1298:
1296:
1115:Parameshvara
928:Parameshvara
698:Dodecahedron
628:
282:Differential
86:
77:
68:
59:
36:
8604:: 335–355.
8331:divided by
8254:28 December
8228:28 December
7814:Woo circles
7794:Ford circle
7721:hypersphere
6572:hypocycloid
6269:cross-ratio
5756:right angle
5155:unit sphere
5144:unit circle
4043:and radius
2358:coordinates
2123:, then the
1634:petroglyphs
1356:Terminology
1240:Present day
1187:Lobachevsky
1174:1700s–1900s
1131:Jyeṣṭhadeva
1123:1400s–1700s
1075:Brahmagupta
898:Lobachevsky
878:Jyeṣṭhadeva
828:Brahmagupta
756:Hypersphere
728:Tetrahedron
703:Icosahedron
275:Diophantine
8807:Categories
8744:PlanetMath
8674:0486406806
8611:2010.12340
8038:References
7937:Of a torus
7121:pseudomath
7113:polynomial
6604:Archimedes
6502:of points
6365:See also:
6060:See also:
5736:See also:
5444:See also:
5066:Properties
3602:(1: −
2557:using the
2241:of radius
2117:arc length
1931:Archimedes
1836:irrational
1680:Book 3 of
1568:metathesis
1500:Semicircle
1366:concentric
1100:al-Yasamin
1044:Apollonius
1039:Archimedes
1029:Pythagoras
1019:Baudhayana
973:al-Yasamin
923:Pythagoras
818:Baudhayana
808:Archimedes
803:Apollonius
708:Octahedron
559:Hypotenuse
434:Similarity
429:Congruence
341:Incidence
292:Symplectic
287:Riemannian
270:Arithmetic
245:Projective
233:Hyperbolic
161:Projecting
8760:MathWorld
8733:EMS Press
8700:. Dover.
8350:1303.0904
8108:130296519
8100:0003-598X
8080:Antiquity
7968:Apeirogon
7642:θ
7639:
7620:θ
7617:
7525:π
7415:⋯
7262:⋯
7132:In other
7082:geometers
7007:−
6998:…
6911:∑
6540:, is any
6514:In every
6486:Thus, if
6136:. By the
6022:collinear
5700:∠
5697:−
5683:∠
5666:∠
5641:⌢
5609:⌢
5024:−
4901:−
4883:−
4867:−
4787:−
4772:−
4747:−
4732:−
4680:−
4645:−
4617:−
4589:−
4417:¯
4401:¯
4390:−
4382:¯
4374:−
4369:¯
4340:−
4292:−
4265:¯
4257:−
4212:for real
4190:¯
4163:¯
4063:−
4007:ϕ
4004:−
4001:θ
3995:
3967:−
3952:±
3946:ϕ
3943:−
3940:θ
3934:
3907:, giving
3883:ϕ
3880:−
3877:θ
3871:
3785:ϕ
3746:θ
3674:ϕ
3671:−
3668:θ
3662:
3640:−
3598:: 0) and
3538:−
3523:−
3446:−
3417:−
3401:−
3385:−
3356:−
3325:−
3296:−
3264:−
3235:−
3197:−
3171:−
3158:−
3142:−
3116:−
3088:−
3062:−
3033:−
3007:−
2730:−
2632:
2596:
2419:−
2394:−
2311:Equations
2273:θ
2245:and with
2213:θ
2184:, or one
2135:θ
2048:≈
2029:π
1970:π
1933:, in his
1886:π
1877:π
1834:(pi), an
1682:Euclid's
1554:The word
1550:Etymology
1376:connected
1346:astronomy
1334:full moon
1217:Minkowski
1136:Descartes
1070:Aryabhata
1065:Kātyāyana
996:by period
908:Minkowski
883:Kātyāyana
843:Descartes
788:Aryabhata
767:Geometers
751:Tesseract
615:Trapezoid
587:Rectangle
380:Dimension
265:Algebraic
255:Synthetic
226:Spherical
211:Euclidean
132:Perimeter
53:A circle
8755:"Circle"
8735:. 2001 .
8723:"Circle"
8624:Archived
8579:12641658
8541:Archived
8522:Archived
8503:Archived
8202:Archived
8183:Archived
8144:7227282M
8124:Archived
8054:Archived
8028:𝜏 (tau)
7957:See also
7846:Incircle
7841:Excircle
7728:topology
7676:for the
7505:, where
7501:using a
7344:‖
7338:‖
7187:‖
7181:‖
7117:rational
6527:vertices
6520:incircle
6516:triangle
6086:constant
5929:midpoint
5723:, where
5576:, where
5432:Theorems
5330:, where
4324:, since
2464:equation
2115:with an
1839:constant
1784:Creation
1726:geometry
1706:Elements
1695:—
1684:Elements
1620:drawing.
1574:κρίκος (
1404:Diameter
1368:circles.
1350:calculus
1323:diameter
1207:Poincaré
1151:Minggatu
1110:Yang Hui
1080:Virasena
968:Yang Hui
963:Virasena
933:Poincaré
913:Minggatu
693:Cylinder
638:Diameter
597:Rhomboid
554:Altitude
545:Triangle
439:Symmetry
417:Parallel
402:Diagonal
372:Features
369:Concepts
260:Analytic
221:Elliptic
203:Branches
189:Timeline
148:Geometry
18:1-Sphere
8813:Circles
8454:4030420
8434:Bibcode
8367:3413900
8174:Proclus
7472:squares
7111:of any
7099:) is a
7079:ancient
6755:figure.
6623:ellipse
5916:compass
5780:versine
5776:sagitta
5762:Sagitta
5565:, then
5463:, then
5419:
5407:
5395:and if
5346:Tangent
5301:, then
5272:, then
5252:If the
5153:of the
5110:similar
5090:is the
4035:In the
3835:. When
3488:, each
2682:,
2674:,
2482:| and |
2332:,
2182:degrees
2167:subtend
1780:compass
1722:science
1692:centre.
1669:
1657:
1594:History
1587:circuit
1570:of the
1506:Tangent
1488:segment
1484:Segment
1362:Annulus
1232:Coxeter
1212:Hilbert
1197:Riemann
1146:Huygens
1105:al-Tusi
1095:Khayyám
1085:Alhazen
1052:1–1400s
953:al-Tusi
938:Riemann
888:Khayyám
873:Huygens
868:Hilbert
838:Coxeter
798:Alhazen
776:by name
713:Pyramid
592:Rhombus
536:Polygon
488:segment
336:Fractal
319:Digital
304:Complex
185:History
180:Outline
137:C = 2πR
8740:Circle
8704:
8671:
8630:17 May
8577:
8452:
8426:Nature
8365:
8310:
8277:
8164:(1959)
8142:
8106:
8098:
8051:krikos
7998:Sphere
7948:
7688:metric
7136:-norms
6977:
6719:, and
6494:, and
6279:, and
6236:is on
6232:where
5228:, the
5224:For a
4499:) to (
4273:
4248:
3725:where
2678:) to (
2654:where
2345:In an
2302:𝜃 = 2
2125:radian
2107:of an
2105:vertex
2095:Radian
2051:0.7854
1937:, the
1702:Book I
1698:Euclid
1615:Arabic
1581:circus
1576:krikos
1556:circle
1518:closed
1494:Secant
1478:Sector
1422:Radius
1398:length
1396:: the
1382:Centre
1374:: any
1319:radius
1315:centre
1307:points
1299:circle
1253:Gromov
1248:Atiyah
1227:Veblen
1222:Cartan
1192:Bolyai
1161:Sakabe
1141:Pascal
1034:Euclid
1024:Manava
958:Veblen
943:Sakabe
918:Pascal
903:Manava
863:Gromov
848:Euclid
833:Cartan
823:Bolyai
813:Atiyah
723:Sphere
686:cuboid
674:Volume
629:Circle
582:Square
500:Length
422:Vertex
326:Convex
309:Finite
250:Affine
165:sphere
83:
74:
65:
56:
42:Circle
8606:arXiv
8575:S2CID
8450:S2CID
8345:arXiv
8104:S2CID
8014:Notes
7169:-norm
7115:with
6626:zero.
6594:with
6500:locus
6111:about
6095:ratio
5380:and ∠
5176:Chord
4463:P = (
4461:. If
2668:angle
2658:is a
2462:This
2237:of a
2109:angle
1714:Plato
1560:Greek
1470:is a
1388:Chord
1342:gears
1338:wheel
1311:plane
1309:in a
1303:shape
1301:is a
1202:Klein
1182:Gauss
1156:Euler
1090:Sijzi
1060:Zhang
1014:Ahmes
978:Zhang
948:Sijzi
893:Klein
858:Gauss
853:Euler
793:Ahmes
526:Plane
461:Point
397:Curve
392:Angle
169:plane
167:to a
8702:ISBN
8669:ISBN
8646:link
8632:2021
8308:ISBN
8275:ISBN
8256:2023
8230:2023
8096:ISSN
7701:and
7109:root
6949:>
6120:and
6101:and
5975:and
5893:/ 2)
5774:The
5623:and
5537:and
5496:and
5488:and
5455:and
5421:arc(
5297:and
5289:and
5268:and
5260:and
5194:The
5184:The
5133:and
5124:area
5122:The
5094:O(2,
4449:The
4433:line
3594:(1:
2186:turn
1860:by:
1788:halo
1778:The
1728:and
1636:and
1628:and
1584:and
1514:open
1416:Lens
1410:Disc
1348:and
1327:disc
1166:Aida
783:Aida
742:Four
681:Cube
648:Area
620:Kite
531:Area
483:Line
120:Area
112:O(2)
95:Type
8742:at
8616:doi
8567:doi
8563:110
8442:doi
8355:doi
8304:108
8088:doi
7744:).
7726:In
7636:cos
7614:sin
7462:In
6738:= 2
6606:to
6371:If
6250:CPD
6242:CPD
6230:BPQ
6224:on
6149:APB
6045:or
5889:= (
5403:DAQ
5387:If
5382:BPA
5378:BOA
5326:− 4
4823:If
4711:or
4530:+ (
3986:sin
3931:cos
3868:cos
3823:= 0
3659:cos
3618:In
3484:In
2842:to
2629:sin
2593:cos
2119:of
2084:).
1747:is
1716:'s
1712:In
1660:256
1372:Arc
1005:BCE
493:ray
8828:Pi
8809::
8794:.
8783:.
8772:.
8757:.
8731:.
8725:.
8642:}}
8638:{{
8614:.
8602:11
8600:.
8596:.
8573:.
8561:.
8473:,
8448:.
8440:.
8430:59
8428:.
8424:.
8363:MR
8361:.
8353:.
8341:46
8339:.
8306:.
8247:.
8221:.
8176:,
8157:,
8140:OL
8102:.
8094:.
8084:87
8082:.
8078:.
7466:,
7160:).
7088:.
7052:.
6750:A
6743:A
6728:=
6715:,
6629:A
6613:A
6570:A
6567:.
6555:A
6532:A
6529:.
6490:,
6377:AB
6339:1.
6275:,
6254:CD
6238:AP
6226:AB
6218:PD
6142:PC
6134:AB
6107:AB
6039:,
6024:).
6007:PR
5992:PQ
5969:,
5881:−
5870:−
5574:√2
5570:=
5552:AD
5550:×
5548:AC
5546:=
5544:AF
5515:AE
5513:×
5511:AB
5509:=
5507:AD
5505:×
5503:AC
5490:AD
5486:AE
5478:AE
5476:×
5474:AB
5472:=
5470:AD
5468:×
5466:AC
5457:EB
5453:CD
5423:AQ
5405:=
5397:AQ
5389:AD
5314:+
5310:+
5306:+
5279:cd
5277:=
5275:ab
5241:).
5221:.
5146:.
5112:.
5075:).
4928:.
4831:≠
4557:,
4545:=
4537:–
4522:−
4506:,
4495:,
4483:,
4470:,
4435:.
4216:,
3843:=
3830:=
3610:.
3572:0.
2986::
2486:−
2478:−
2376:,
2364:,
2188:.
2091:.
2001:,
1811:.
1704:,
1700:,
1677:.
1666:81
1648:.
1646:Bi
1352:.
1329:.
1297:A
163:a
125:πR
8798:.
8763:.
8746:.
8710:.
8677:.
8648:)
8634:.
8618::
8608::
8581:.
8569::
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8436::
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8357::
8347::
8333:d
8329:C
8316:.
8283:.
8258:.
8232:.
8110:.
8090::
8030:.
7737:R
7706:∞
7703:L
7699:1
7696:L
7692:r
7686:∞
7683:L
7680:(
7674:r
7646:|
7632:|
7628:+
7624:|
7610:|
7605:1
7600:=
7597:r
7577:1
7574:=
7570:|
7566:y
7562:|
7558:+
7554:|
7550:x
7546:|
7515:r
7511:r
7507:r
7489:r
7484:2
7468:p
7448:.
7441:2
7436:|
7431:n
7427:x
7423:|
7418:+
7412:+
7407:2
7402:|
7397:2
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7384:+
7379:2
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7369:1
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7361:|
7354:=
7349:2
7341:x
7326:p
7312:.
7307:p
7303:/
7299:1
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7288:p
7283:|
7278:n
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7270:|
7265:+
7259:+
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7244:2
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7216:1
7212:x
7208:|
7202:(
7197:=
7192:p
7184:x
7167:p
7157:p
7151:p
7134:p
7097:π
7038:n
7034:P
7013:;
7010:1
7004:n
7001:,
6995:,
6992:2
6989:,
6986:1
6983:=
6980:m
6968:,
6963:m
6960:2
6956:R
6952:n
6944:m
6941:2
6936:i
6932:d
6926:n
6921:1
6918:=
6915:i
6890:R
6868:i
6864:d
6843:m
6840:2
6818:n
6814:P
6793:n
6772:n
6740:.
6736:n
6730:a
6726:b
6721:n
6717:b
6713:a
6699:1
6696:=
6690:n
6685:|
6680:b
6677:y
6672:|
6667:+
6661:n
6656:|
6651:a
6648:x
6643:|
6610:.
6600:n
6596:n
6504:P
6496:C
6492:B
6488:A
6467:|
6463:C
6460:B
6456:|
6449:|
6445:C
6442:A
6438:|
6431:=
6424:|
6420:P
6417:B
6413:|
6406:|
6402:P
6399:A
6395:|
6381:P
6373:C
6355:P
6336:=
6331:|
6326:]
6323:P
6320:,
6317:C
6314:;
6311:B
6308:,
6305:A
6302:[
6297:|
6285:P
6281:C
6277:B
6273:A
6246:P
6234:Q
6222:D
6202:.
6196:C
6193:B
6188:C
6185:A
6179:=
6173:P
6170:B
6165:P
6162:A
6130:C
6126:P
6122:B
6118:A
6103:B
6099:A
6083:2
6080:d
6078:/
6076:1
6073:d
6048:R
6042:Q
6036:P
6030:M
6017:M
6011:.
5996:.
5981:,
5978:R
5972:Q
5966:P
5942:M
5933:M
5897:r
5891:y
5887:x
5885:)
5883:x
5879:r
5877:2
5872:x
5868:r
5866:2
5862:x
5858:y
5842:.
5837:2
5834:x
5829:+
5823:x
5820:8
5814:2
5810:y
5804:=
5801:r
5791:x
5787:y
5725:O
5710:C
5707:O
5704:B
5693:E
5690:O
5687:D
5680:=
5676:B
5673:A
5670:C
5663:2
5637:C
5634:B
5605:E
5602:D
5589:A
5582:r
5578:ℓ
5572:r
5568:ℓ
5563:°
5539:D
5535:C
5531:A
5527:F
5523:A
5498:C
5494:B
5481:.
5461:A
5427:.
5425:)
5416:2
5413:/
5410:1
5401:∠
5393:A
5374:O
5370:P
5366:B
5362:A
5336:p
5332:r
5328:p
5324:r
5316:d
5312:c
5308:b
5304:a
5299:d
5295:c
5291:b
5287:a
5282:.
5270:d
5266:c
5262:b
5258:a
5171:.
5161:.
5135:π
5131:π
5119:.
5105:.
5103:T
5096:R
5051:.
5044:1
5040:y
5034:1
5030:x
5021:=
5015:x
5012:d
5007:y
5004:d
4981:,
4976:2
4972:r
4968:=
4965:y
4960:1
4956:y
4952:+
4949:x
4944:1
4940:x
4910:.
4904:b
4896:1
4892:y
4886:a
4878:1
4874:x
4864:=
4858:x
4855:d
4850:y
4847:d
4833:b
4829:1
4826:y
4809:.
4804:2
4800:r
4796:=
4793:)
4790:b
4784:y
4781:(
4778:)
4775:b
4767:1
4763:y
4759:(
4756:+
4753:)
4750:a
4744:x
4741:(
4738:)
4735:a
4727:1
4723:x
4719:(
4699:,
4694:1
4690:y
4686:)
4683:b
4675:1
4671:y
4667:(
4664:+
4659:1
4655:x
4651:)
4648:a
4640:1
4636:x
4632:(
4629:=
4626:y
4623:)
4620:b
4612:1
4608:y
4604:(
4601:+
4598:x
4595:)
4592:a
4584:1
4580:x
4576:(
4566:c
4562:1
4559:y
4555:1
4552:x
4547:c
4543:y
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4539:b
4535:1
4532:y
4528:x
4526:)
4524:a
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4517:x
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4511:1
4508:y
4504:1
4501:x
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4493:a
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4455:P
4414:c
4409:c
4406:+
4398:z
4393:c
4387:z
4379:c
4366:z
4361:z
4358:=
4353:2
4348:|
4343:c
4337:z
4333:|
4310:2
4305:|
4300:c
4296:|
4287:2
4283:r
4279:=
4276:q
4270:,
4262:c
4254:=
4251:g
4245:,
4242:1
4239:=
4236:p
4222:g
4218:q
4214:p
4198:q
4195:=
4186:z
4183:g
4177:+
4174:z
4171:g
4168:+
4160:z
4155:z
4152:p
4130:.
4127:c
4124:+
4119:t
4116:i
4112:e
4108:r
4105:=
4102:z
4080:.
4077:r
4074:=
4070:|
4066:c
4060:z
4056:|
4045:r
4041:c
4015:.
4010:)
3998:(
3990:2
3980:2
3975:0
3971:r
3962:2
3958:a
3949:)
3937:(
3926:0
3922:r
3918:=
3915:r
3905:r
3889:.
3886:)
3874:(
3865:a
3862:2
3859:=
3856:r
3845:a
3841:0
3838:r
3832:a
3828:r
3821:0
3818:r
3813:x
3809:φ
3805:0
3802:r
3788:)
3782:,
3777:0
3773:r
3769:(
3749:)
3743:,
3740:r
3737:(
3727:a
3711:,
3706:2
3702:a
3698:=
3693:2
3688:0
3684:r
3680:+
3677:)
3665:(
3654:0
3650:r
3646:r
3643:2
3635:2
3631:r
3604:i
3600:J
3596:i
3592:I
3569:=
3564:2
3560:z
3556:c
3553:+
3550:z
3547:y
3544:b
3541:2
3535:z
3532:x
3529:a
3526:2
3518:2
3514:y
3510:+
3505:2
3501:x
3465:.
3459:)
3454:1
3450:x
3441:3
3437:x
3433:(
3430:)
3425:2
3421:y
3412:3
3408:y
3404:(
3398:)
3393:2
3389:x
3380:3
3376:x
3372:(
3369:)
3364:1
3360:y
3351:3
3347:y
3343:(
3338:)
3333:2
3329:y
3320:3
3316:y
3312:(
3309:)
3304:1
3300:y
3291:3
3287:y
3283:(
3280:+
3277:)
3272:2
3268:x
3259:3
3255:x
3251:(
3248:)
3243:1
3239:x
3230:3
3226:x
3222:(
3216:=
3210:)
3205:1
3201:x
3192:x
3187:(
3184:)
3179:2
3175:y
3166:y
3161:(
3155:)
3150:2
3146:x
3137:x
3132:(
3129:)
3124:1
3120:y
3111:y
3106:(
3101:)
3096:2
3092:y
3083:y
3078:(
3075:)
3070:1
3066:y
3057:y
3052:(
3049:+
3046:)
3041:2
3037:x
3028:x
3023:(
3020:)
3015:1
3011:x
3002:x
2997:(
2968:)
2963:3
2959:y
2955:,
2950:3
2946:x
2942:(
2939:,
2936:)
2931:2
2927:y
2923:,
2918:2
2914:x
2910:(
2907:,
2904:)
2899:1
2895:y
2891:,
2886:1
2882:x
2878:(
2860:t
2852:x
2844:r
2840:t
2820:.
2812:2
2808:t
2804:+
2801:1
2796:t
2793:2
2787:r
2784:+
2781:b
2778:=
2771:y
2764:,
2756:2
2752:t
2748:+
2745:1
2738:2
2734:t
2727:1
2721:r
2718:+
2715:a
2712:=
2705:x
2688:x
2684:y
2680:x
2676:b
2672:a
2664:π
2656:t
2638:,
2635:t
2625:r
2622:+
2619:b
2616:=
2609:y
2602:,
2599:t
2589:r
2586:+
2583:a
2580:=
2573:x
2534:.
2529:2
2525:r
2521:=
2516:2
2512:y
2508:+
2503:2
2499:x
2488:b
2484:y
2480:a
2476:x
2448:.
2443:2
2439:r
2435:=
2430:2
2426:)
2422:b
2416:y
2413:(
2410:+
2405:2
2401:)
2397:a
2391:x
2388:(
2378:y
2374:x
2370:r
2366:b
2362:a
2360:(
2351:y
2349:–
2347:x
2334:b
2330:a
2326:r
2304:π
2286:.
2281:2
2277:r
2268:2
2265:1
2260:=
2257:A
2243:r
2235:A
2219:,
2216:r
2210:=
2207:s
2197:r
2193:s
2177:π
2175:2
2151:.
2146:r
2143:s
2138:=
2121:s
2101:r
2082:d
2064:,
2059:2
2055:d
2043:4
2037:2
2033:d
2023:=
2019:a
2016:e
2013:r
2010:A
1999:d
1983:.
1978:2
1974:r
1967:=
1963:a
1960:e
1957:r
1954:A
1943:π
1917:π
1892:.
1889:d
1883:=
1880:r
1874:2
1871:=
1868:C
1858:d
1854:r
1850:C
1845:π
1843:2
1832:π
1790:.
1745:π
1674:π
1663:/
1458:0
1455:=
1452:r
1432:r
1286:e
1279:t
1272:v
413:)
409:(
191:)
187:(
87:O
78:R
69:D
60:C
34:.
20:)
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