2453:
1903:
3158:
1022:
2525:
2448:{\displaystyle {\begin{aligned}T&={\tfrac {1}{2}}{\begin{vmatrix}x_{A}&x_{B}&x_{C}\\y_{A}&y_{B}&y_{C}\\1&1&1\end{vmatrix}}={\tfrac {1}{2}}{\begin{vmatrix}x_{A}&x_{B}\\y_{A}&y_{B}\end{vmatrix}}+{\tfrac {1}{2}}{\begin{vmatrix}x_{B}&x_{C}\\y_{B}&y_{C}\end{vmatrix}}+{\tfrac {1}{2}}{\begin{vmatrix}x_{C}&x_{A}\\y_{C}&y_{A}\end{vmatrix}}\\&={\tfrac {1}{2}}(x_{A}y_{B}-x_{B}y_{A}+x_{B}y_{C}-x_{C}y_{B}+x_{C}y_{A}-x_{A}y_{C}),\end{aligned}}}
3798:
1687:
526:
329:
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343:
448:
49:
1118:
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623:
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3149:
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675:: the object can be balanced on its centroid in a uniform gravitational field. The centroid cuts every median in the ratio 2:1, i.e. the distance between a vertex and the centroid is twice the distance between the centroid and the midpoint of the opposite side. If one reflects a median in the angle bisector that passes through the same vertex, one obtains a
2509:
states that the sum of the lengths of any two sides of a triangle must be greater than or equal to the length of the third side. A triangle with three given positive side lengths exists if and only if those side lengths satisfy the triangle inequality. The sum of two side lengths can equal the length
2536:
from pressure to one of its points, triangles are sturdy because specifying the lengths of all three sides determines the angles. Therefore, a triangle will not change shape unless its sides are bent or extended or broken or if its joints break; in essence, each of the three sides supports the other
1025:
This diagram illustrates the geometric principle of angle-angle-side triangle congruence: given triangle ABC and triangle A'B'C', triangle ABC is congruent with triangle A'B'C' if and only if: angle CAB is congruent with angle C'A'B', and angle ABC is congruent with angle A'B'C', and BC is congruent
4010:
The triangles in both spaces have properties different from the triangles in
Euclidean space. For example, as mentioned above, the internal angles of a triangle in Euclidean space always add up to 180°. However, the sum of the internal angles of a hyperbolic triangle is less than 180°, and for any
1073:
have exactly the same size and shape. All pairs of congruent triangles are also similar, but not all pairs of similar triangles are congruent. Given two congruent triangles, all pairs of corresponding interior angles are equal in measure, and all pairs of corresponding sides have the same length.
427:
Each triangle has many special points inside it, on its edges, or otherwise associated with it. They are constructed by finding three lines associated symmetrically with the three sides (or vertices) and then proving that the three lines meet in a single point. An important tool for proving the
3202:
Every acute triangle has three inscribed squares (squares in its interior such that all four of a square's vertices lie on a side of the triangle, so two of them lie on the same side and hence one side of the square coincides with part of a side of the triangle). In a right triangle two of the
614:(red line). The center of the nine-point circle lies at the midpoint between the orthocenter and the circumcenter, and the distance between the centroid and the circumcenter is half that between the centroid and the orthocenter. Generally, the incircle's center is not located on Euler's line.
493:
implies that if the circumcenter is located on the side of the triangle, then the angle opposite that side is a right angle. If the circumcenter is located inside the triangle, then the triangle is acute; if the circumcenter is located outside the triangle, then the triangle is obtuse.
3134:{\displaystyle {\frac {{\overline {PA}}\cdot {\overline {QA}}}{{\overline {CA}}\cdot {\overline {AB}}}}+{\frac {{\overline {PB}}\cdot {\overline {QB}}}{{\overline {AB}}\cdot {\overline {BC}}}}+{\frac {{\overline {PC}}\cdot {\overline {QC}}}{{\overline {BC}}\cdot {\overline {CA}}}}=1.}
280:
A triangle is a figure consisting of three line segments, each of whose endpoints are connected. This forms a polygon with three sides and three angles. The terminology for categorizing triangles is more than two thousand years old, having been defined in Book One of Euclid's
3207:
inscribed squares. An obtuse triangle has only one inscribed square, with a side coinciding with part of the triangle's longest side. Within a given triangle, a longer common side is associated with a smaller inscribed square. If an inscribed square has a side of length
1040:, if every angle of one triangle has the same measure as the corresponding angle in the other triangle. The corresponding sides of similar triangles have lengths that are in the same proportion, and this property is also sufficient to establish similarity.
1012:
2690:
Two systems avoid that feature, so that the coordinates of a point are not affected by moving the triangle, rotating it, or reflecting it as in a mirror, any of which gives a congruent triangle, or even by rescaling it to a similar triangle:
3755:
As mentioned above, every triangle has a unique circumcircle, a circle passing through all three vertices, whose center is the intersection of the perpendicular bisectors of the triangle's sides. Furthermore, every triangle has a unique
4134:
The definition by Euclid states that an isosceles triangle is a triangle with exactly two equal sides. By the modern definition, it has at least two equal sides, implying that an equilateral triangle is a special case of isosceles
1057:
If and only if one pair of corresponding sides of two triangles are in the same proportion as another pair of corresponding sides, and their included angles have the same measure, then the triangles are similar. (The
3476:
3183:
of that point. If the interior point is the circumcenter of the reference triangle, the vertices of the pedal triangle are the midpoints of the reference triangle's sides, and so the pedal triangle is called the
6203:
Advances in
Discrete and Computational Geometry: Proceedings of the 1996 AMS-IMS-SIAM Joint Summer Research Conference, Discrete and Computational Geometry—Ten Years Later, July 14-18, 1996, Mount Holyoke
4011:
spherical triangle, the sum is more than 180°. In particular, it is possible to draw a triangle on a sphere such that the measure of each of its internal angles equals 90°, adding up to a total of 270°. By
755:
Any three angles that add to 180° can be the internal angles of a triangle. Infinitely many triangles have the same angles, since specifying the angles of a triangle does not determine its size. (A
2687:, and to use Cartesian coordinates. While convenient for many purposes, this approach has the disadvantage of all points' coordinate values being dependent on the arbitrary placement in the plane.
1908:
889:
1475:
729:. The sum of the measures of the three exterior angles (one for each vertex) of any triangle is 360 degrees, and indeed, this is true for any convex polygon, no matter how many sides it has.
1682:
2589:, a polygon is subdivided into multiple triangles that are attached edge-to-edge, with the property that their vertices coincide with the set of vertices of the polygon. In the case of a
4061:
1098:
AAS: Two angles and a corresponding (non-included) side in a triangle have the same measure and length, respectively, as those in the other triangle. (This is sometimes referred to as
1600:
2792:
1243:
1153:, which has area 1. There are several ways to calculate the area of an arbitrary triangle. One of the oldest and simplest is to take half the product of the length of one side
3760:, which passes through the triangle's vertices and has its center at the triangle's centroid. Of all ellipses going through the triangle's vertices, it has the smallest area.
1420:
725:) to an interior angle. The measure of an exterior angle of a triangle is equal to the sum of the measures of the two interior angles that are not adjacent to it; this is the
2794:
specify the point's location by the relative weights that would have to be put on the three vertices in order to balance the otherwise weightless triangle on the given point.
3621:
1890:
1840:
1790:
2808:
As discussed above, every triangle has a unique inscribed circle (incircle) that is interior to the triangle and tangent to all three sides. Every triangle has a unique
4611:
It is well known that the incenter of a
Euclidean triangle lies on its Euler line connecting the centroid and the circumcenter if and only if the triangle is isosceles.
3512:
2485:
1088:
ASA: Two interior angles and the side between them in a triangle have the same measure and length, respectively, as those in the other triangle. (This is the basis of
1054:
one pair of internal angles of two triangles have the same measure as each other, and another pair also have the same measure as each other, the triangles are similar.
1377:
881:
841:
821:
781:
861:
801:
3546:
3327:
3280:
3233:
2728:
3920:
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260:
Triangles are classified into different types based on their angles and the lengths of their sides. Relations between angles and side lengths are a major focus of
2848:
2754:
2665:
2639:
608:. The orthocenter (blue point), the center of the nine-point circle (red), the centroid (orange), and the circumcenter (green) all lie on a single line, known as
3698:
744:. In a scalene triangle, the trigonometric functions can be used to find the unknown measure of either a side or an internal angle; methods for doing so use the
4090:
relating the properties of a triangle in the space to properties of a corresponding triangle in a model space like hyperbolic or elliptic space. For example, a
4081:
3862:
3675:
3586:
3566:
3367:
3347:
3300:
3253:
2888:
2868:
2824:
of a triangle is the ellipse inscribed within the triangle tangent to its sides at the contact points of its excircles. For any ellipse inscribed in a triangle
2612:
1544:
1524:
1504:
1355:
1331:
1291:
1267:
1197:
1173:
1149:
3821:, which can be made by intersecting three circles of equal size. The construction may be performed with a compass alone without needing a straightedge, by the
3372:
1721:
can be defined without reference to a notion of distance or squares. In any affine space (including
Euclidean planes), every triangle with the same base and
3836:
subset of the plane lying between three mutually tangent convex regions. These sides are three smoothed curved lines connecting their endpoints called the
1247:
This formula can be proven by cutting up the triangle and an identical copy into pieces and rearranging the pieces into the shape of a rectangle of base
823:, each of them between 0° and 180°, to be the angles of a triangle can also be stated using trigonometric functions. For example, a triangle with angles
511:
of the altitude. The length of the altitude is the distance between the base and the vertex. The three altitudes intersect in a single point, called the
562:
of a triangle is a straight line through a vertex that cuts the corresponding angle in half. The three angle bisectors intersect in a single point, the
3188:
or medial triangle. The midpoint triangle subdivides the reference triangle into four congruent triangles which are similar to the reference triangle.
3642:
with vertices given by the six intersections of the sides of a triangle with the three lines that are parallel to the sides and that pass through its
584:; they lie outside the triangle and touch one side, as well as the extensions of the other two. The centers of the incircles and excircles form an
477:
of the side and being perpendicular to it, forming a right angle with it. The three perpendicular bisectors meet in a single point, the triangle's
4847:
6340:
Algorithmic
Foundation of Robotics VII: Selected Contributions of the Seventh International Workshop on the Algorithmic Foundations of Robotics
1085:
SAS Postulate: Two sides in a triangle have the same length as two sides in the other triangle, and the included angles have the same measure.
6388:
6355:
6264:
6239:
6212:
6144:
6051:
5969:
5895:
5828:
5807:
5785:
5760:
5735:
5676:
5652:
5633:
5614:
5487:
5427:
2759:
2683:
One way to identify locations of points in (or outside) a triangle is to place the triangle in an arbitrary location and orientation in the
763:, has internal angles of 0° and 180°; whether such a shape counts as a triangle is a matter of convention.) The conditions for three angles
5534:
550:
is a straight line through the orthocenter (blue), the center of the nine-point circle (red), centroid (orange), and circumcenter (green).
5727:
3199:
of a reference triangle has its vertices at the points of tangency of the reference triangle's excircles with its sides (not extended).
6453:
4567:
3813:
edges. The edges of a circular triangle may be either convex (bending outward) or concave (bending inward). The intersection of three
1427:
1121:
The area formula for a triangle can be proven by cutting two copies of the triangle into pieces and rearranging them into a rectangle.
1078:
6425:
6324:
6302:
6283:
6086:
6021:
5865:
5579:
5509:
196:
1609:
665:
of the opposite side, and divides the triangle into two equal areas. The three medians intersect in a single point, the triangle's
290:
Triangles have many types based on the length of the sides and on the angles. A triangle whose sides are all the same length is an
5879:
3195:
of a reference triangle has its vertices at the three points of tangency of the reference triangle's sides with its incircle. The
1065:
If and only if three pairs of corresponding sides of two triangles are all in the same proportion, then the triangles are similar.
6200:
Pocchiola, Michel; Vegter, Gert (1999). "On
Polygonal Covers". In Chazelle, Bernard; Goodman, Jacob E.; Pollack, Richard (eds.).
422:
6185:
Oxman, Victor; Stupel, Moshe (2013). "Why Are the Side
Lengths of the Squares Inscribed in a Triangle so Close to Each Other?".
3778:, one with maximal area can be found in linear time; its vertices may be chosen as three of the vertices of the given polygon.
436:. Similarly, lines associated with a triangle are often constructed by proving that three symmetrically constructed points are
287:. The names used for modern classification are either a direct transliteration of Euclid's Greek or their Latin translations.
594:. The remaining three points for which it is named are the midpoints of the portion of altitude between the vertices and the
3588:. The smallest possible ratio of the side of one inscribed square to the side of another in the same non-obtuse triangle is
671:
or geometric barycenter. The centroid of a rigid triangular object (cut out of a thin sheet of uniform density) is also its
444:
gives a useful general criterion. In this section, just a few of the most commonly encountered constructions are explained.
5980:
5040:
1729:
with the same base whose opposite side lies on the parallel line. This affine approach was developed in Book 1 of Euclid's
503:
of a triangle is a straight line through a vertex and perpendicular to the opposite side. This opposite side is called the
6417:
6013:
5985:
5907:
1204:
699:
The measures of the interior angles of the triangle always add up to 180 degrees (same color to point out they are equal).
5212:
Eddy, R. H.; Fritsch, R. (1994). "The Conics of Ludwig
Kiepert: A Comprehensive Lesson in the Geometry of the Triangle".
5662:
3822:
715:. This allows the determination of the measure of the third angle of any triangle, given the measure of two angles. An
455:
is the center of a circle passing through the three vertices of the triangle; the intersection of the altitudes is the
6576:
6556:
6412:
704:
139:
3817:
forms a circular triangle whose sides are all convex. An example of a circular triangle with three convex edges is a
3478:
The largest possible ratio of the area of the inscribed square to the area of the triangle is 1/2, which occurs when
5417:
6551:
6508:
6483:
5752:
3179:
From an interior point in a reference triangle, the nearest points on the three sides serve as the vertices of the
588:. The midpoints of the three sides and the feet of the three altitudes all lie on a single circle, the triangle's
2730:
indicate that the ratio of the distance of the point from the first side to its distance from the second side is
1089:
574:. The incircle is the circle that lies inside the triangle and touches all three sides. Its radius is called the
328:
31:
6611:
6156:
4852:
4018:
2582:
756:
1549:
2671:, a vertex connected by two other vertices, the diagonal between which lies entirely within the polygon. The
342:
6536:
5961:
2765:
1007:{\displaystyle \cos ^{2}\alpha +\cos ^{2}\beta +\cos ^{2}\gamma +2\cos(\alpha )\cos(\beta )\cos(\gamma )=1.}
265:
5447:
Allaire, Patricia R.; Zhou, Junmin; Yao, Haishen (2012). "Proving a nineteenth century ellipse identity"".
1725:
has its apex (the third vertex) on a line parallel to the base, and their common area is half of that of a
257:
sides. This article is about straight-sided triangles in
Euclidean geometry, except where otherwise noted.
6561:
6446:
5916:
4585:
4157:
if, given any two points in that subset, the whole line segment joining them also lies within that subset.
4106:
3840:. Any pseudotriangle can be partitioned into many pseudotriangles with the boundaries of convex disks and
3833:
1021:
733:
726:
207:
6962:
6902:
6541:
6365:
Verdiyan, Vardan; Salas, Daniel Campos (2007). "Simple trigonometric substitutions with broad results".
3736:
3157:
2695:
2586:
2569:
1741:
1714:
1070:
1036:
600:. The radius of the nine-point circle is half that of the circumcircle. It touches the incircle (at the
356:
227:
5686:
Chandran, Sharat; Mount, David M. (1992). "A parallel algorithm for enclosed and enclosing triangles".
2524:
507:
of the altitude, and the point where the altitude intersects the base (or its extension) is called the
6061:
Longuet-Higgins, Michael S. (2003). "On the ratio of the inradius to the circumradius of a triangle".
4583:
Edmonds, Allan L.; Hajja, Mowaffaq; Martini, Horst (2008). "Orthocentric simplices and biregularity".
3983:, roughly speaking a flat space. This means triangles may also be discovered in several spaces, as in
3797:
2537:
two. A rectangle, in contrast, is more dependent on the strength of its joints in a structural sense.
1384:
398:
6846:
6616:
6546:
6488:
6256:
6063:
5567:
5546:
5526:
5449:
441:
334:
291:
283:
3591:
2820:. This ellipse has the greatest area of any ellipse tangent to all three sides of the triangle. The
6952:
6927:
6897:
6892:
6851:
6566:
6187:
5921:
5838:
4838:
4012:
3992:
3967:
3935:
3745:
3203:
squares coincide and have a vertex at the triangle's right angle, so a right triangle has only two
2813:
2519:
2506:
2500:
1849:
1799:
1749:
1706:
1686:
1483:
722:
585:
499:
384:
370:
219:
4265:
3650:, the Lemoine hexagon is interior to the triangle with two vertices on each side of the triangle.
6983:
6957:
6498:
6173:
6103:
5957:
Mathematical
Techniques: An Introduction for the Engineering, Physical, and Mathematical Sciences
5934:
5719:
5167:
4087:
4000:
3988:
3971:
3939:
3814:
2488:
1737:
1112:
1095:
SSS: Each side of a triangle has the same length as the corresponding side of the other triangle.
712:
348:
295:
215:
211:
180:
173:
1479:
517:
of the triangle. The orthocenter lies inside the triangle if and only if the triangle is acute.
490:
70:
4842:
6937:
6531:
6439:
6407:
6384:
6351:
6320:
6298:
6279:
6260:
6235:
6208:
6140:
6047:
6017:
5965:
5891:
5824:
5803:
5781:
5756:
5731:
5672:
5648:
5629:
5610:
5575:
5530:
5505:
5483:
5423:
5331:
5296:
5060:
4916:
4726:
4642:
4626:
4563:
4543:
4524:
4508:
4492:
4446:
4414:
4398:
4382:
4366:
3818:
3806:
3792:
3764:
3481:
3185:
3171:
2821:
2809:
2668:
2460:
657:
651:
601:
590:
543:
429:
250:
223:
168:
113:
60:
5312:
5280:
5257:
4988:
4824:
4710:
4682:
4474:
4465:
4430:
4336:
4327:
4300:
4281:
4247:
3748:
of a reference triangle (other than a right triangle) is the triangle whose sides are on the
1362:
866:
826:
806:
766:
6466:
6374:
6343:
6227:
6165:
6132:
6095:
6072:
6039:
5994:
5926:
5883:
5861:
5857:
5848:
5715:
5695:
5602:
5555:
5475:
5458:
5221:
5194:
5159:
5052:
4594:
4557:
4238:
4211:
4180:
3984:
3196:
3192:
2672:
1895:
846:
786:
433:
192:
121:
6115:
5946:
5707:
5624:
Berg, Mark Theodoor de; Kreveld, Marc van; Overmars, Mark H.; Schwarzkopf, Otfried (2000).
5589:
5345:
5079:
4606:
3517:
3305:
3258:
3211:
2701:
6932:
6912:
6907:
6877:
6596:
6571:
6503:
6111:
5942:
5703:
5585:
4602:
3980:
3896:
3867:
3757:
3643:
3635:
2684:
2574:
2510:
of the third side only in the case of a degenerate triangle, one with collinear vertices.
741:
737:
708:
683:
525:
404:
315:
162:
156:
126:
56:
5905:
Hungerbühler, Norbert (1994). "A short elementary proof of the Mohr-Mascheroni theorem".
3825:. Alternatively, it can be constructed by rounding the sides of an equilateral triangle.
2827:
2733:
2644:
2618:
2585:
means the partition of any planar object into a collection of triangles. For example, in
610:
547:
3680:
6942:
6922:
6887:
6882:
6513:
6493:
6312:
5497:
4110:
4066:
3996:
3847:
3829:
3810:
3775:
3771:
that passes through the triangle's three vertices, its centroid, and its circumcenter.
3660:
3654:
3571:
3551:
3352:
3332:
3285:
3238:
3180:
3167:
2873:
2853:
2597:
2590:
1894:
for the vertices of a triangle, its relative oriented area can be calculated using the
1529:
1509:
1489:
1340:
1316:
1276:
1252:
1182:
1158:
1134:
1074:
This is a total of six equalities, but three are often sufficient to prove congruence.
1051:
884:
749:
717:
672:
558:
390:
376:
311:
307:
269:
143:
135:
5544:
Bailey, Herbert; Detemple, Duane (1998). "Squares inscribed in angles and triangles".
3958:
2675:
states that every simple polygon that is not itself a triangle has at least two ears.
1117:
447:
48:
6977:
6917:
6768:
6661:
6581:
6523:
5666:
5185:
Silvester, John R. (March 2017). "Extremal area ellipses of a convex quadrilateral".
3701:
2533:
1726:
1603:
184:
151:
732:
Another relation between the internal angles and triangles creates a new concept of
695:
6947:
6817:
6773:
6737:
6727:
6722:
5998:
5795:
5559:
5225:
4091:
3949:
3749:
3717:
2541:
1718:
760:
745:
485:
479:
452:
261:
254:
188:
147:
131:
5371:
534:
6378:
6335:
6250:
6201:
6124:
6007:
5955:
5873:
5818:
5771:
5746:
5520:
6856:
6763:
6742:
6732:
3923:
2557:
1722:
1705:
is a line (dashed green) parallel to the base. This is the Euclidean version of
1486:, is a formula for finding the area of a triangle from the lengths of its sides
1298:
631:
622:
596:
513:
456:
303:
200:
5852:
6861:
6717:
6707:
6591:
6347:
6231:
6136:
6076:
6043:
6031:
5887:
5699:
5606:
5479:
5462:
5163:
5150:
Smith, Geoff; Leversha, Gerry (November 2007). "Euler and triangle geometry".
5056:
4598:
4154:
3995:, and it can be obtained by drawing on a negatively curved surface, such as a
3726:
3626:
2698:
specify the relative distances of a point from the sides, so that coordinates
2553:
1713:
Because the ratios between areas of shapes in the same plane are preserved by
1062:
for any two sides of a polygon is the internal angle between those two sides.)
1027:
5842:
5064:
4003:, and it can be obtained by drawing on a positively curved surface such as a
2812:
which is interior to the triangle and tangent at the midpoints of the sides.
640:
The incircle of a triangle, and the intersection of the medians known as the
6836:
6826:
6803:
6793:
6783:
6712:
6621:
6586:
5777:
3841:
3148:
677:
469:
437:
314:, and a triangle in which one of it angles is greater than that angle is an
118:
6338:. In Akella, Srinivas; Amato, Nancy M.; Huang, Wesley; Mishra, Bud (eds.).
5372:"LAS 100 — Freshman Seminar — Fall 1996: Reasoning with shape and quantity"
6154:
Oldknow, Adrian (1995). "Computer Aided Research into Triangle Geometry".
546:
demonstrates a symmetry where six points lie on the edge of the triangle.
183:, any two points determine a unique line segment situated within a unique
17:
6841:
6831:
6788:
6747:
6676:
6666:
667:
662:
641:
605:
580:
570:
564:
474:
460:
232:
108:
5171:
4559:
Geometry Turned On: Dynamic Software in Learning, Teaching, and Research
3471:{\displaystyle q_{a}={\frac {2Ta}{a^{2}+2T}}={\frac {ah_{a}}{a+h_{a}}}.}
6798:
6778:
6691:
6686:
6681:
6671:
6646:
6601:
6462:
6177:
6107:
5938:
5198:
4102:
3647:
3639:
2817:
2667:
diagonals. Triangulation of a simple polygon has a relationship to the
2545:
1044:
310:, a triangle in which all of its angles are less than that angle is an
100:
3623:. Both of these extreme cases occur for the isosceles right triangle.
6606:
6009:
Geometry Transformed: Euclidean Plane Geometry Based on Rigid Motions
4004:
2549:
319:
30:
This article is about the basic geometric shape. For other uses, see
6169:
6099:
5930:
432:, which gives a criterion for determining when three such lines are
230:
enclosed by three sides which are straight relative to the surface (
1379:
are known, then the altitude can be calculated using trigonometry,
6651:
6431:
6428:. Lists some 5200 interesting points associated with any triangle.
5820:
Structure and Form in Design: Critical Ideas for Creative Practice
5688:
International Journal of Computational Geometry & Applications
3796:
3768:
3625:
2573:
2523:
1685:
1297:
1116:
1020:
694:
446:
246:
104:
6319:. Research in Mathematics Education. Information Age Publishing.
2540:
Triangles are strong in terms of rigidity, but while packed in a
6125:"On Geodesic Triangles with Right Angles in a Dually Flat Space"
1126:
80:
6435:
5574:. DMV Seminar 25. Basel: Birkhäuser Verlag. pp. viii+112.
4083:
is the fraction of the sphere's area enclosed by the triangle.
3255:, part of which side coincides with a side of the square, then
473:
of a side of a triangle is a straight line passing through the
459:. The intersection of the angle bisectors is the center of the
4940:
2552:). Tessellated triangles still maintain superior strength for
2548:
under compression (hence the prevalence of hexagonal forms in
3828:
A special case of concave circular triangle can be seen in a
3801:
Circular triangles with a mixture of convex and concave edges
740:, as well as the other functions. They can be defined as the
6317:
The Classification of Quadrilaterals: A Study in Definition
5628:(2 ed.). Berlin Heidelberg: Springer. pp. 45–61.
3548:, and the altitude of the triangle from the base of length
721:
of a triangle is an angle that is a linear pair (and hence
711:
is always 180 degrees. This fact is equivalent to Euclid's
5847:. Vol. 1 (2nd ed.). Cambridge University Press.
3752:
to the reference triangle's circumcircle at its vertices.
191:
determine a unique triangle situated within a unique flat
6129:
Progress in Information Geometry: Theory and Applications
4562:. The Mathematical Association of America. pp. 3–4.
3999:. Likewise, a triangle in spherical geometry is called a
294:, a triangle with two sides having the same length is an
3677:
can be inscribed in a triangle of area at most equal to
705:
sum of the measures of the interior angles of a triangle
681:. The three symmedians intersect in a single point, the
298:, and a triangle with three different-length sides is a
3735:
The circumscribed circle tangent to a triangle and the
176:
equals one-half the product of height and base length.
5645:
Problems in Euclidean Space: Applications of Convexity
5323:
5321:
2286:
2216:
2200:
2137:
2121:
2058:
2042:
1938:
1922:
1560:
1438:
1215:
1129:
is defined by comparison with a square of side length
5730:. Vol. 19. Mathematical Association of America.
5276:
4144:
Again, in all cases "mirror images" are also similar.
4069:
4021:
4015:, the sum of the angles of a triangle on a sphere is
3899:
3870:
3850:
3683:
3663:
3594:
3574:
3554:
3520:
3484:
3375:
3355:
3335:
3308:
3288:
3261:
3241:
3214:
2896:
2876:
2856:
2830:
2768:
2736:
2704:
2647:
2621:
2600:
2463:
1906:
1852:
1802:
1752:
1612:
1552:
1532:
1512:
1492:
1430:
1387:
1365:
1343:
1319:
1279:
1255:
1207:
1185:
1161:
1137:
892:
869:
849:
829:
809:
789:
769:
417:
Points, lines, and circles associated with a triangle
103:
with three corners and three sides, one of the basic
5346:"The area of a spherical triangle. Girard's Theorem"
5308:
5080:"Reflection-Induced Perspectivities Among Triangles"
3979:
A non-planar triangle is a triangle not included in
166:; the shortest segment between base and apex is the
138:, each one bounded by a pair of adjacent edges; the
6870:
6816:
6756:
6700:
6639:
6630:
6522:
6474:
6334:Vahedi, Mostafa; van der Stappen, A. Frank (2008).
5626:
Computational geometry: algorithms and applications
4535:
4533:
79:
69:
55:
41:
6131:. Signals and Communication Technology. Springer.
4075:
4055:
3914:
3885:
3856:
3692:
3669:
3615:
3580:
3560:
3540:
3506:
3470:
3361:
3341:
3321:
3294:
3274:
3247:
3227:
3133:
2882:
2862:
2842:
2786:
2748:
2722:
2659:
2633:
2606:
2479:
2447:
1884:
1834:
1784:
1676:
1594:
1538:
1518:
1498:
1469:
1414:
1371:
1349:
1325:
1285:
1261:
1237:
1191:
1167:
1143:
1006:
875:
855:
835:
815:
795:
775:
160:, in which case the opposite vertex is called the
154:. Sometimes an arbitrary edge is chosen to be the
2556:, however, which is why engineering makes use of
489:, the circle passing through all three vertices.
1470:{\displaystyle T={\tfrac {1}{2}}ab\sin \gamma .}
1030:are used here to show angle and side equalities.
5292:
5026:
4746:
4678:
3864:disks in a pseudotriangle, the partition gives
3844:, a process known as pseudo-triangulation. For
742:ratio between any two sides of a right triangle
578:. There are three other important circles, the
6084:Meisters, G. H. (1975). "Polygons have ears".
5745:Devadoss, Satyan L.; O'Rourke, Joseph (2011).
5106:
5014:
4305:
146:(180 degrees or π radians). The triangle is a
6447:
5668:The First Six Books of the Elements of Euclid
5416:Frame, Michael; Urry, Amelia (21 June 2016).
5238:
4741:
3991:. A triangle in hyperbolic space is called a
1677:{\displaystyle T={\sqrt {s(s-a)(s-b)(s-c)}}.}
1081:for a pair of triangles to be congruent are:
302:. A triangle in which one of the angles is a
124:while the sides connecting them, also called
8:
6336:"Caging Polygons with Two and Three Fingers"
6222:Ramsay, Arlan; Richtmyer, Robert D. (1995).
5262:
5041:"Twenty-one points on the nine-point circle"
4556:Schattschneider, Doris; King, James (1997).
4426:
2532:Unlike a rectangle, which may collapse into
1177:(the base) times the corresponding altitude
214:) also determine a triangle, for instance a
5572:Lectures on spaces of nonpositive curvature
5125:
5111:
4912:
3630:The Lemoine hexagon inscribed in a triangle
2544:arrangement triangles are not as strong as
1302:Applying trigonometry to find the altitude
655:of a triangle is a straight line through a
6636:
6454:
6440:
6432:
5419:Fractal Worlds: Grown, Built, and Imagined
5078:Moses, Peter; Kimberling, Charles (2009).
4461:
4410:
4394:
4323:
4234:
4207:
4176:
3700:. Equality holds only if the polygon is a
736:. The primary trigonometric functions are
318:. These definitions date back at least to
5981:"An Elementary Proof of Marden's Theorem"
5920:
5137:
4068:
4056:{\displaystyle 180^{\circ }\times (1+4f)}
4026:
4020:
3898:
3869:
3849:
3682:
3662:
3648:simple form or its self-intersecting form
3605:
3598:
3593:
3573:
3553:
3530:
3519:
3489:
3483:
3456:
3438:
3428:
3407:
3389:
3380:
3374:
3354:
3334:
3313:
3307:
3287:
3266:
3260:
3240:
3219:
3213:
3107:
3089:
3072:
3054:
3051:
3030:
3012:
2995:
2977:
2974:
2953:
2935:
2918:
2900:
2897:
2895:
2875:
2855:
2829:
2767:
2735:
2703:
2646:
2620:
2599:
2472:
2464:
2462:
2429:
2419:
2406:
2396:
2383:
2373:
2360:
2350:
2337:
2327:
2314:
2304:
2285:
2261:
2249:
2235:
2223:
2211:
2199:
2182:
2170:
2156:
2144:
2132:
2120:
2103:
2091:
2077:
2065:
2053:
2041:
2007:
1995:
1983:
1969:
1957:
1945:
1933:
1921:
1907:
1905:
1873:
1860:
1851:
1823:
1810:
1801:
1773:
1760:
1751:
1717:, the relative areas of triangles in any
1619:
1611:
1559:
1551:
1531:
1511:
1491:
1437:
1429:
1386:
1364:
1342:
1318:
1278:
1254:
1214:
1206:
1184:
1160:
1136:
935:
916:
897:
891:
868:
848:
828:
808:
788:
768:
226:is a region of a general two-dimensional
210:, three "straight" segments (having zero
5647:. Dover Publications. pp. 149–160.
5403:
4952:
1595:{\displaystyle s={\tfrac {1}{2}}(a+b+c)}
568:, which is the center of the triangle's
6383:(4th ed.). John Wiley & Sons.
5844:The Thirteen Books of Euclid's Elements
5599:A panoramic view of Riemannian geometry
5327:
4984:
4979:
4964:
4885:
4848:MacTutor History of Mathematics Archive
4169:
4127:
366:
324:
5391:
5002:
4928:
4879:
4808:
4796:
4784:
4772:
4760:
4694:
4666:
4105:shapes based on triangles include the
4094:is characterized by such comparisons.
3774:Of all triangles contained in a given
3708:Figures circumscribed about a triangle
3235:and the triangle has a side of length
2787:{\displaystyle \alpha :\beta :\gamma }
38:
5422:. Yale University Press. p. 21.
4900:
4864:
4722:
4706:
4654:
4638:
4378:
4362:
4350:
4277:
4261:
4216:
4192:
3926:of any pseudotriangle is a triangle.
7:
5671:(facsimile ed.). TASCHEN GmbH.
5253:
4820:
4622:
4539:
4520:
4504:
4488:
4470:
4442:
4332:
4296:
4243:
1238:{\displaystyle T={\tfrac {1}{2}}bh.}
6224:Introduction to Hyperbolic Geometry
5748:Discrete and Computational Geometry
5728:Anneli Lax New Mathematical Library
1079:necessary and sufficient conditions
266:sine, cosine, and tangent functions
187:, and any three points that do not
5470:Anglin, W. S.; Lambek, J. (1995).
5344:Polking, John C. (25 April 1999).
4086:In more general spaces, there are
1701:and area. The locus of their apex
1424:, so the area of the triangle is:
483:; this point is the center of the
276:Definition, terminology, and types
268:relate side lengths and angles in
25:
6087:The American Mathematical Monthly
5954:Jordan, D. W.; Smith, P. (2010).
5277:Vahedi & van der Stappen 2008
5087:Journal for Geometry and Graphics
2578:Triangulation in a simple polygon
2528:Rigidity of a triangle and square
197:three-dimensional Euclidean space
195:. More generally, four points in
189:all lie on the same straight line
6426:Encyclopedia of triangle centers
5039:Kimberling, Clark (March 2008).
3957:
3948:
3725:
3716:
3156:
3147:
2641:triangles that are separated by
1415:{\displaystyle h=a\sin(\gamma )}
630:
621:
533:
524:
423:Encyclopedia of Triangle Centers
397:
383:
369:
355:
341:
327:
47:
5875:Geometry: Our Cultural Heritage
4353:, Definition 20, Definition 21.
2804:Figures inscribed in a triangle
6036:Geometry: A High School Course
5999:10.1080/00029890.2008.11920532
5560:10.1080/0025570X.1998.11996652
5226:10.1080/0025570X.1994.11996212
4050:
4035:
3616:{\displaystyle 2{\sqrt {2}}/3}
2473:
2465:
2435:
2297:
1879:
1853:
1829:
1803:
1779:
1753:
1666:
1654:
1651:
1639:
1636:
1624:
1589:
1571:
1409:
1403:
995:
989:
980:
974:
965:
959:
1:
6295:Geometry from Euclid to Knots
6207:. American Mathematical Soc.
6014:American Mathematical Society
5986:American Mathematical Monthly
5908:American Mathematical Monthly
5800:The Cartoon Guide to Geometry
5773:Real-Time Collision Detection
1885:{\displaystyle (x_{C},y_{C})}
1835:{\displaystyle (x_{B},y_{B})}
1785:{\displaystyle (x_{A},y_{A})}
1102:and then includes ASA above.)
1047:about similar triangles are:
1034:Two triangles are said to be
428:existence of these points is
6315:; Griffin, Jennifer (2008).
5309:Devadoss & O'Rourke 2011
5015:Allaire, Zhou & Yao 2012
3930:Triangle in non-planar space
3809:is a triangle with circular
3117:
3099:
3082:
3064:
3040:
3022:
3005:
2987:
2963:
2945:
2928:
2910:
6413:Encyclopedia of Mathematics
6127:. In Nielsen, Frank (ed.).
5293:Pocchiola & Vegter 1999
5027:Coxeter & Greitzer 1967
4679:Ramsay & Richtmyer 1995
140:sum of angles of a triangle
111:. The corners, also called
7000:
6038:(2nd ed.). Springer.
5770:Ericson, Christer (2005).
5753:Princeton University Press
5643:Eggleston, H. G. (2007) .
5107:Bailey & Detemple 1998
4811:, pp. 65, 72–73, 111.
4306:Usiskin & Griffin 2008
3933:
3790:
3349:, and the triangle's area
2567:
2564:Triangulation of a polygon
2517:
2498:
1110:
1090:surveying by triangulation
420:
29:
6348:10.1007/978-3-540-68405-3
6232:10.1007/978-1-4757-5585-5
6137:10.1007/978-3-030-65459-7
6077:10.1017/S0025557200172249
6044:10.1007/978-1-4757-2022-8
5888:10.1007/978-3-642-14441-7
5700:10.1142/S0218195992000123
5607:10.1007/978-3-642-18245-7
5480:10.1007/978-1-4612-0803-7
5463:10.1017/S0025557200004277
5239:Chandran & Mount 1992
5164:10.1017/S0025557200182087
5057:10.1017/S002555720018249X
4742:Verdiyan & Salas 2007
4599:10.1007/s00025-008-0294-4
1359:and their included angle
1017:Similarity and congruence
87:
46:
32:Triangle (disambiguation)
6367:Mathematical Reflections
6274:Smith, James T. (2000).
6157:The Mathematical Gazette
5522:Algebra and Trigonometry
5187:The Mathematical Gazette
5045:The Mathematical Gazette
4853:University of St Andrews
4427:Anglin & Lambek 1995
3832:. A pseudotriangle is a
3507:{\displaystyle a^{2}=2T}
3369:are related according to
2480:{\displaystyle |\cdot |}
1125:In the Euclidean plane,
208:non-Euclidean geometries
6123:Nielsen, Frank (2021).
6034:; Murrow, Gene (1988).
6006:King, James R. (2021).
5962:Oxford University Press
5597:Berger, Marcel (2002).
5519:Axler, Sheldon (2012).
5126:Oxman & Stupel 2013
5112:Oxman & Stupel 2013
4913:Jordan & Smith 2010
4153:A subset of a plane is
3823:Mohr–Mascheroni theorem
3782:Miscellaneous triangles
2760:Barycentric coordinates
1372:{\displaystyle \gamma }
1069:Two triangles that are
876:{\displaystyle \gamma }
836:{\displaystyle \alpha }
816:{\displaystyle \gamma }
776:{\displaystyle \alpha }
734:trigonometric functions
134:. A triangle has three
6406:Ivanov, A.B. (2001) .
5817:Hann, Michael (2014).
5472:The Heritage of Thales
5350:Geometry of the Sphere
4586:Results in Mathematics
4462:Lang & Murrow 1988
4411:Lang & Murrow 1988
4395:Lang & Murrow 1988
4324:Lang & Murrow 1988
4235:Lang & Murrow 1988
4208:Lang & Murrow 1988
4177:Lang & Murrow 1988
4077:
4057:
3916:
3887:
3858:
3802:
3694:
3671:
3631:
3617:
3582:
3562:
3542:
3508:
3472:
3363:
3343:
3323:
3296:
3276:
3249:
3229:
3135:
2884:
2864:
2844:
2816:shows how to find the
2788:
2750:
2724:
2661:
2635:
2608:
2579:
2529:
2481:
2449:
1886:
1836:
1786:
1715:affine transformations
1710:
1678:
1596:
1540:
1520:
1500:
1471:
1416:
1373:
1351:
1327:
1308:
1287:
1263:
1239:
1193:
1169:
1145:
1122:
1031:
1008:
877:
857:
856:{\displaystyle \beta }
837:
817:
797:
796:{\displaystyle \beta }
777:
727:exterior angle theorem
700:
470:perpendicular bisector
464:
249:sides, for instance a
245:is a shape with three
150:and its interior is a
130:, are one-dimensional
27:Shape with three sides
6257:John Wiley & Sons
5527:John Wiley & Sons
5029:, pp. 18, 23–25.
4867:, Propositions 36–41.
4843:"Heron of Alexandria"
4266:p. 187, Definition 20
4078:
4058:
3922:bitangent lines. The
3917:
3888:
3859:
3800:
3758:Steiner circumellipse
3737:Steiner circumellipse
3695:
3672:
3629:
3618:
3583:
3563:
3543:
3541:{\displaystyle q=a/2}
3509:
3473:
3364:
3344:
3324:
3322:{\displaystyle h_{a}}
3297:
3277:
3275:{\displaystyle q_{a}}
3250:
3230:
3228:{\displaystyle q_{a}}
3136:
2885:
2865:
2845:
2789:
2751:
2725:
2723:{\displaystyle x:y:z}
2696:Trilinear coordinates
2662:
2636:
2609:
2587:polygon triangulation
2577:
2570:Polygon triangulation
2527:
2482:
2450:
1887:
1837:
1787:
1742:Cartesian coordinates
1689:
1679:
1597:
1541:
1521:
1501:
1472:
1417:
1374:
1352:
1328:
1301:
1288:
1264:
1240:
1194:
1170:
1146:
1120:
1024:
1009:
878:
858:
838:
818:
798:
778:
759:, whose vertices are
698:
450:
264:. In particular, the
75:{3} (for equilateral)
6687:Nonagon/Enneagon (9)
6617:Tangential trapezoid
6293:Stahl, Saul (2003).
6252:Geometry For Dummies
6064:Mathematical Gazette
5979:Kalman, Dan (2008).
5547:Mathematics Magazine
5450:Mathematical Gazette
5214:Mathematics Magazine
5152:Mathematical Gazette
4839:Robertson, Edmund F.
4747:Longuet-Higgins 2003
4067:
4019:
3915:{\displaystyle 3n-3}
3897:
3893:pseudotriangles and
3886:{\displaystyle 2n-2}
3868:
3848:
3681:
3661:
3592:
3572:
3552:
3518:
3482:
3373:
3353:
3333:
3306:
3286:
3259:
3239:
3212:
2894:
2874:
2854:
2828:
2818:foci of this ellipse
2766:
2734:
2702:
2645:
2619:
2598:
2461:
1904:
1850:
1800:
1750:
1610:
1550:
1530:
1510:
1490:
1428:
1385:
1363:
1341:
1317:
1277:
1253:
1205:
1183:
1159:
1135:
890:
867:
847:
827:
807:
787:
767:
335:Equilateral triangle
292:equilateral triangle
6799:Megagon (1,000,000)
6567:Isosceles trapezoid
6276:Methods of Geometry
6249:Ryan, Mark (2008).
6188:Forum Geometricorum
5858:Dover reprint, 1956
5853:2027/uva.x001426155
5406:, p. viii+112.
5394:, pp. 134–139.
5140:, pp. 149–160.
4837:O'Connor, John J.;
4763:, pp. 157–167.
4697:, pp. 224–225.
4669:, pp. 107–109.
4088:comparison theorems
3993:hyperbolic triangle
3968:Hyperbolic triangle
3936:Hyperbolic triangle
3746:tangential triangle
2843:{\displaystyle ABC}
2749:{\displaystyle x:y}
2679:Location of a point
2660:{\displaystyle n-3}
2634:{\displaystyle n-2}
2558:tetrahedral trusses
2520:Structural rigidity
2507:triangle inequality
2501:Triangle inequality
1484:Heron of Alexandria
757:degenerate triangle
586:orthocentric system
220:hyperbolic triangle
6769:Icositetragon (24)
6424:Clark Kimberling:
6164:(485 =): 263–274.
5872:Holme, A. (2010).
5802:. William Morrow.
5724:Geometry Revisited
5199:10.1017/mag.2017.2
4195:, pp. xx–xxi.
4073:
4053:
4001:spherical triangle
3989:spherical geometry
3972:spherical triangle
3940:Spherical triangle
3912:
3883:
3854:
3803:
3787:Circular triangles
3693:{\displaystyle 2T}
3690:
3667:
3632:
3613:
3578:
3558:
3538:
3504:
3468:
3359:
3339:
3319:
3292:
3272:
3245:
3225:
3131:
2880:
2860:
2850:, let the foci be
2840:
2784:
2746:
2720:
2657:
2631:
2604:
2580:
2530:
2489:matrix determinant
2477:
2445:
2443:
2295:
2269:
2209:
2190:
2130:
2111:
2051:
2032:
1931:
1882:
1832:
1782:
1738:affine coordinates
1711:
1674:
1592:
1569:
1536:
1516:
1496:
1467:
1447:
1412:
1369:
1347:
1323:
1309:
1283:
1259:
1235:
1224:
1189:
1165:
1141:
1123:
1113:Area of a triangle
1077:Some individually
1032:
1004:
873:
853:
833:
813:
793:
773:
713:parallel postulate
701:
465:
349:Isosceles triangle
296:isosceles triangle
216:spherical triangle
181:Euclidean geometry
174:area of a triangle
6971:
6970:
6812:
6811:
6789:Myriagon (10,000)
6774:Triacontagon (30)
6738:Heptadecagon (17)
6728:Pentadecagon (15)
6723:Tetradecagon (14)
6662:Quadrilateral (4)
6532:Antiparallelogram
6390:978-1-119-32113-2
6357:978-3-540-68405-3
6297:. Prentice-Hall.
6266:978-0-470-08946-0
6241:978-1-4757-5585-5
6214:978-0-8218-0674-6
6146:978-3-030-65458-0
6053:978-1-4757-2022-8
5971:978-0-19-928201-2
5897:978-3-642-14441-7
5830:978-1-4725-8431-1
5823:. A&C Black.
5809:978-0-06-315757-6
5787:978-1-55860-732-3
5762:978-0-691-14553-2
5737:978-0-88385-619-2
5716:Coxeter, H. S. M.
5678:978-3-8365-4471-9
5654:978-0-486-45846-5
5635:978-3-540-65620-3
5616:978-3-642-18245-7
5489:978-1-4612-0803-7
5429:978-0-300-22070-4
5263:Hungerbühler 1994
4657:, Proposition 32.
4107:Sierpiński gasket
4076:{\displaystyle f}
3857:{\displaystyle n}
3819:Reuleaux triangle
3807:circular triangle
3793:Circular triangle
3765:Kiepert hyperbola
3670:{\displaystyle T}
3603:
3581:{\displaystyle a}
3561:{\displaystyle a}
3463:
3423:
3362:{\displaystyle T}
3342:{\displaystyle a}
3295:{\displaystyle a}
3248:{\displaystyle a}
3186:midpoint triangle
3172:Gergonne triangle
3123:
3120:
3102:
3085:
3067:
3046:
3043:
3025:
3008:
2990:
2969:
2966:
2948:
2931:
2913:
2883:{\displaystyle Q}
2863:{\displaystyle P}
2822:Mandart inellipse
2810:Steiner inellipse
2615:sides, there are
2607:{\displaystyle n}
2294:
2208:
2129:
2050:
1930:
1690:Orange triangles
1669:
1568:
1539:{\displaystyle c}
1519:{\displaystyle b}
1499:{\displaystyle a}
1446:
1350:{\displaystyle b}
1326:{\displaystyle a}
1286:{\displaystyle h}
1262:{\displaystyle b}
1223:
1192:{\displaystyle h}
1168:{\displaystyle b}
1144:{\displaystyle 1}
687:of the triangle.
591:nine-point circle
544:Nine-point circle
442:Menelaus' theorem
251:circular triangle
224:geodesic triangle
93:
92:
16:(Redirected from
6991:
6784:Chiliagon (1000)
6764:Icositrigon (23)
6743:Octadecagon (18)
6733:Hexadecagon (16)
6637:
6456:
6449:
6442:
6433:
6421:
6394:
6370:
6361:
6330:
6308:
6289:
6270:
6245:
6218:
6196:
6181:
6150:
6119:
6080:
6057:
6027:
6002:
5975:
5960:(4th ed.).
5950:
5924:
5901:
5856:
5839:Heath, Thomas L.
5834:
5813:
5791:
5766:
5741:
5711:
5682:
5658:
5639:
5620:
5593:
5568:Ballmann, Werner
5563:
5540:
5536:978-0470-58579-5
5515:
5493:
5466:
5434:
5433:
5413:
5407:
5401:
5395:
5389:
5383:
5382:
5380:
5378:
5367:
5361:
5360:
5358:
5356:
5341:
5335:
5325:
5316:
5306:
5300:
5290:
5284:
5274:
5268:
5248:
5242:
5236:
5230:
5229:
5209:
5203:
5202:
5182:
5176:
5175:
5158:(522): 436–452.
5147:
5141:
5135:
5129:
5123:
5117:
5101:
5095:
5094:
5084:
5075:
5069:
5068:
5036:
5030:
5024:
5018:
5012:
5006:
5000:
4994:
4974:
4968:
4962:
4956:
4950:
4944:
4941:Berg et al. 2000
4938:
4932:
4926:
4920:
4910:
4904:
4903:, p. 86–87.
4898:
4892:
4874:
4868:
4862:
4856:
4855:
4834:
4828:
4818:
4812:
4806:
4800:
4794:
4788:
4782:
4776:
4770:
4764:
4758:
4752:
4736:
4730:
4720:
4714:
4704:
4698:
4692:
4686:
4676:
4670:
4664:
4658:
4652:
4646:
4636:
4630:
4620:
4614:
4613:
4580:
4574:
4573:
4553:
4547:
4537:
4528:
4518:
4512:
4502:
4496:
4486:
4480:
4456:
4450:
4440:
4434:
4424:
4418:
4408:
4402:
4392:
4386:
4376:
4370:
4360:
4354:
4348:
4342:
4318:
4312:
4291:
4285:
4275:
4269:
4259:
4253:
4229:
4223:
4202:
4196:
4190:
4184:
4174:
4158:
4151:
4145:
4142:
4136:
4132:
4098:Fractal geometry
4082:
4080:
4079:
4074:
4062:
4060:
4059:
4054:
4031:
4030:
4013:Girard's theorem
3985:hyperbolic space
3961:
3952:
3921:
3919:
3918:
3913:
3892:
3890:
3889:
3884:
3863:
3861:
3860:
3855:
3834:simply-connected
3729:
3720:
3699:
3697:
3696:
3691:
3676:
3674:
3673:
3668:
3646:. In either its
3622:
3620:
3619:
3614:
3609:
3604:
3599:
3587:
3585:
3584:
3579:
3567:
3565:
3564:
3559:
3547:
3545:
3544:
3539:
3534:
3513:
3511:
3510:
3505:
3494:
3493:
3477:
3475:
3474:
3469:
3464:
3462:
3461:
3460:
3444:
3443:
3442:
3429:
3424:
3422:
3412:
3411:
3401:
3390:
3385:
3384:
3368:
3366:
3365:
3360:
3348:
3346:
3345:
3340:
3328:
3326:
3325:
3320:
3318:
3317:
3301:
3299:
3298:
3293:
3281:
3279:
3278:
3273:
3271:
3270:
3254:
3252:
3251:
3246:
3234:
3232:
3231:
3226:
3224:
3223:
3197:extouch triangle
3193:intouch triangle
3160:
3151:
3140:
3138:
3137:
3132:
3124:
3122:
3121:
3116:
3108:
3103:
3098:
3090:
3087:
3086:
3081:
3073:
3068:
3063:
3055:
3052:
3047:
3045:
3044:
3039:
3031:
3026:
3021:
3013:
3010:
3009:
3004:
2996:
2991:
2986:
2978:
2975:
2970:
2968:
2967:
2962:
2954:
2949:
2944:
2936:
2933:
2932:
2927:
2919:
2914:
2909:
2901:
2898:
2889:
2887:
2886:
2881:
2869:
2867:
2866:
2861:
2849:
2847:
2846:
2841:
2814:Marden's theorem
2793:
2791:
2790:
2785:
2755:
2753:
2752:
2747:
2729:
2727:
2726:
2721:
2673:two ears theorem
2666:
2664:
2663:
2658:
2640:
2638:
2637:
2632:
2614:
2613:
2611:
2610:
2605:
2486:
2484:
2483:
2478:
2476:
2468:
2454:
2452:
2451:
2446:
2444:
2434:
2433:
2424:
2423:
2411:
2410:
2401:
2400:
2388:
2387:
2378:
2377:
2365:
2364:
2355:
2354:
2342:
2341:
2332:
2331:
2319:
2318:
2309:
2308:
2296:
2287:
2278:
2274:
2273:
2266:
2265:
2254:
2253:
2240:
2239:
2228:
2227:
2210:
2201:
2195:
2194:
2187:
2186:
2175:
2174:
2161:
2160:
2149:
2148:
2131:
2122:
2116:
2115:
2108:
2107:
2096:
2095:
2082:
2081:
2070:
2069:
2052:
2043:
2037:
2036:
2012:
2011:
2000:
1999:
1988:
1987:
1974:
1973:
1962:
1961:
1950:
1949:
1932:
1923:
1896:shoelace formula
1893:
1891:
1889:
1888:
1883:
1878:
1877:
1865:
1864:
1843:
1841:
1839:
1838:
1833:
1828:
1827:
1815:
1814:
1793:
1791:
1789:
1788:
1783:
1778:
1777:
1765:
1764:
1707:Lexell's theorem
1704:
1700:
1696:
1683:
1681:
1680:
1675:
1670:
1620:
1601:
1599:
1598:
1593:
1570:
1561:
1545:
1543:
1542:
1537:
1525:
1523:
1522:
1517:
1505:
1503:
1502:
1497:
1476:
1474:
1473:
1468:
1448:
1439:
1423:
1421:
1419:
1418:
1413:
1378:
1376:
1375:
1370:
1358:
1356:
1354:
1353:
1348:
1334:
1332:
1330:
1329:
1324:
1307:
1294:
1292:
1290:
1289:
1284:
1270:
1268:
1266:
1265:
1260:
1244:
1242:
1241:
1236:
1225:
1216:
1200:
1198:
1196:
1195:
1190:
1176:
1174:
1172:
1171:
1166:
1152:
1150:
1148:
1147:
1142:
1026:with B'C'. Note
1013:
1011:
1010:
1005:
940:
939:
921:
920:
902:
901:
882:
880:
879:
874:
862:
860:
859:
854:
842:
840:
839:
834:
822:
820:
819:
814:
802:
800:
799:
794:
782:
780:
779:
774:
634:
625:
604:) and the three
537:
528:
401:
387:
373:
362:Scalene triangle
359:
345:
331:
300:scalene triangle
242:
241:
142:always equals a
85:various methods;
51:
39:
21:
6999:
6998:
6994:
6993:
6992:
6990:
6989:
6988:
6974:
6973:
6972:
6967:
6866:
6820:
6808:
6752:
6718:Tridecagon (13)
6708:Hendecagon (11)
6696:
6632:
6626:
6597:Right trapezoid
6518:
6470:
6460:
6405:
6402:
6397:
6391:
6373:
6364:
6358:
6333:
6327:
6313:Usiskin, Zalman
6311:
6305:
6292:
6286:
6273:
6267:
6248:
6242:
6221:
6215:
6199:
6184:
6170:10.2307/3618298
6153:
6147:
6122:
6100:10.2307/2319703
6083:
6060:
6054:
6030:
6024:
6005:
5978:
5972:
5953:
5931:10.2307/2974536
5904:
5898:
5871:
5837:
5831:
5816:
5810:
5794:
5788:
5769:
5763:
5744:
5738:
5720:Greitzer, S. L.
5714:
5685:
5679:
5661:
5655:
5642:
5636:
5623:
5617:
5596:
5582:
5566:
5543:
5537:
5518:
5512:
5498:Apostol, Tom M.
5496:
5490:
5469:
5446:
5442:
5437:
5430:
5415:
5414:
5410:
5402:
5398:
5390:
5386:
5376:
5374:
5369:
5368:
5364:
5354:
5352:
5343:
5342:
5338:
5326:
5319:
5307:
5303:
5291:
5287:
5275:
5271:
5267:
5249:
5245:
5237:
5233:
5211:
5210:
5206:
5184:
5183:
5179:
5149:
5148:
5144:
5136:
5132:
5124:
5120:
5116:
5102:
5098:
5082:
5077:
5076:
5072:
5038:
5037:
5033:
5025:
5021:
5013:
5009:
5001:
4997:
4993:
4975:
4971:
4963:
4959:
4951:
4947:
4939:
4935:
4927:
4923:
4911:
4907:
4899:
4895:
4891:
4888:, p. 34–35
4875:
4871:
4863:
4859:
4836:
4835:
4831:
4819:
4815:
4807:
4803:
4795:
4791:
4783:
4779:
4771:
4767:
4759:
4755:
4751:
4737:
4733:
4721:
4717:
4705:
4701:
4693:
4689:
4677:
4673:
4665:
4661:
4653:
4649:
4637:
4633:
4621:
4617:
4582:
4581:
4577:
4570:
4555:
4554:
4550:
4538:
4531:
4519:
4515:
4503:
4499:
4487:
4483:
4479:
4457:
4453:
4441:
4437:
4425:
4421:
4409:
4405:
4393:
4389:
4377:
4373:
4361:
4357:
4349:
4345:
4341:
4319:
4315:
4311:
4292:
4288:
4276:
4272:
4260:
4256:
4252:
4230:
4226:
4222:
4219:, Definition 20
4203:
4199:
4191:
4187:
4175:
4171:
4167:
4162:
4161:
4152:
4148:
4143:
4139:
4133:
4129:
4124:
4119:
4100:
4065:
4064:
4022:
4017:
4016:
3981:Euclidean space
3977:
3976:
3975:
3974:
3964:
3963:
3962:
3954:
3953:
3942:
3934:Main articles:
3932:
3895:
3894:
3866:
3865:
3846:
3845:
3842:bitangent lines
3795:
3789:
3784:
3742:
3741:
3740:
3739:
3732:
3731:
3730:
3722:
3721:
3710:
3679:
3678:
3659:
3658:
3644:symmedian point
3636:Lemoine hexagon
3590:
3589:
3570:
3569:
3550:
3549:
3516:
3515:
3485:
3480:
3479:
3452:
3445:
3434:
3430:
3403:
3402:
3391:
3376:
3371:
3370:
3351:
3350:
3331:
3330:
3309:
3304:
3303:
3284:
3283:
3262:
3257:
3256:
3237:
3236:
3215:
3210:
3209:
3177:
3176:
3175:
3174:
3163:
3162:
3161:
3153:
3152:
3109:
3091:
3088:
3074:
3056:
3053:
3032:
3014:
3011:
2997:
2979:
2976:
2955:
2937:
2934:
2920:
2902:
2899:
2892:
2891:
2872:
2871:
2852:
2851:
2826:
2825:
2806:
2801:
2799:Related figures
2764:
2763:
2732:
2731:
2700:
2699:
2685:Cartesian plane
2681:
2643:
2642:
2617:
2616:
2596:
2595:
2594:
2572:
2566:
2522:
2516:
2503:
2497:
2459:
2458:
2442:
2441:
2425:
2415:
2402:
2392:
2379:
2369:
2356:
2346:
2333:
2323:
2310:
2300:
2276:
2275:
2268:
2267:
2257:
2255:
2245:
2242:
2241:
2231:
2229:
2219:
2212:
2189:
2188:
2178:
2176:
2166:
2163:
2162:
2152:
2150:
2140:
2133:
2110:
2109:
2099:
2097:
2087:
2084:
2083:
2073:
2071:
2061:
2054:
2031:
2030:
2025:
2020:
2014:
2013:
2003:
2001:
1991:
1989:
1979:
1976:
1975:
1965:
1963:
1953:
1951:
1941:
1934:
1914:
1902:
1901:
1869:
1856:
1848:
1847:
1845:
1819:
1806:
1798:
1797:
1795:
1769:
1756:
1748:
1747:
1745:
1702:
1698:
1691:
1608:
1607:
1548:
1547:
1528:
1527:
1508:
1507:
1488:
1487:
1480:Heron's formula
1426:
1425:
1383:
1382:
1380:
1361:
1360:
1339:
1338:
1336:
1315:
1314:
1312:
1303:
1275:
1274:
1272:
1251:
1250:
1248:
1203:
1202:
1181:
1180:
1178:
1157:
1156:
1154:
1133:
1132:
1130:
1115:
1109:
1019:
931:
912:
893:
888:
887:
865:
864:
845:
844:
825:
824:
805:
804:
785:
784:
765:
764:
738:sine and cosine
709:Euclidean space
693:
684:symmedian point
647:
646:
645:
644:
637:
636:
635:
627:
626:
602:Feuerbach point
554:
553:
552:
551:
540:
539:
538:
530:
529:
491:Thales' theorem
425:
419:
414:
407:
405:Obtuse triangle
402:
393:
388:
379:
374:
363:
360:
351:
346:
337:
332:
316:obtuse triangle
278:
270:right triangles
239:
238:
136:internal angles
86:
71:Schläfli symbol
35:
28:
23:
22:
15:
12:
11:
5:
6997:
6995:
6987:
6986:
6976:
6975:
6969:
6968:
6966:
6965:
6960:
6955:
6950:
6945:
6940:
6935:
6930:
6925:
6923:Pseudotriangle
6920:
6915:
6910:
6905:
6900:
6895:
6890:
6885:
6880:
6874:
6872:
6868:
6867:
6865:
6864:
6859:
6854:
6849:
6844:
6839:
6834:
6829:
6823:
6821:
6814:
6813:
6810:
6809:
6807:
6806:
6801:
6796:
6791:
6786:
6781:
6776:
6771:
6766:
6760:
6758:
6754:
6753:
6751:
6750:
6745:
6740:
6735:
6730:
6725:
6720:
6715:
6713:Dodecagon (12)
6710:
6704:
6702:
6698:
6697:
6695:
6694:
6689:
6684:
6679:
6674:
6669:
6664:
6659:
6654:
6649:
6643:
6641:
6634:
6628:
6627:
6625:
6624:
6619:
6614:
6609:
6604:
6599:
6594:
6589:
6584:
6579:
6574:
6569:
6564:
6559:
6554:
6549:
6544:
6539:
6534:
6528:
6526:
6524:Quadrilaterals
6520:
6519:
6517:
6516:
6511:
6506:
6501:
6496:
6491:
6486:
6480:
6478:
6472:
6471:
6461:
6459:
6458:
6451:
6444:
6436:
6430:
6429:
6422:
6401:
6400:External links
6398:
6396:
6395:
6389:
6375:Young, Cynthia
6371:
6362:
6356:
6331:
6325:
6309:
6303:
6290:
6284:
6271:
6265:
6246:
6240:
6219:
6213:
6197:
6182:
6151:
6145:
6120:
6094:(6): 648–651.
6081:
6058:
6052:
6028:
6022:
6003:
5993:(4): 330–338.
5976:
5970:
5951:
5922:10.1.1.45.9902
5915:(8): 784–787.
5902:
5896:
5869:
5835:
5829:
5814:
5808:
5792:
5786:
5767:
5761:
5742:
5736:
5712:
5694:(2): 191–214.
5683:
5677:
5659:
5653:
5640:
5634:
5621:
5615:
5594:
5580:
5564:
5554:(4): 278–284.
5541:
5535:
5516:
5510:
5502:Linear Algebra
5494:
5488:
5467:
5443:
5441:
5438:
5436:
5435:
5428:
5408:
5396:
5384:
5362:
5336:
5317:
5301:
5285:
5269:
5266:
5265:
5260:
5250:
5243:
5231:
5220:(3): 188–205.
5204:
5193:(550): 11–26.
5177:
5142:
5138:Eggleston 2007
5130:
5118:
5115:
5114:
5109:
5103:
5096:
5070:
5051:(523): 29–38.
5031:
5019:
5007:
4995:
4992:
4991:
4982:
4976:
4969:
4957:
4945:
4933:
4931:, p. 125.
4921:
4905:
4893:
4890:
4889:
4883:
4876:
4869:
4857:
4829:
4813:
4801:
4789:
4787:, p. 171.
4777:
4775:, p. 167.
4765:
4753:
4750:
4749:
4744:
4738:
4731:
4715:
4699:
4687:
4671:
4659:
4647:
4631:
4615:
4593:(1–2): 41–50.
4575:
4569:978-0883850992
4568:
4548:
4529:
4513:
4497:
4481:
4478:
4477:
4468:
4458:
4451:
4435:
4419:
4403:
4387:
4371:
4355:
4343:
4340:
4339:
4330:
4320:
4313:
4310:
4309:
4303:
4293:
4286:
4270:
4254:
4251:
4250:
4241:
4231:
4224:
4221:
4220:
4214:
4204:
4197:
4185:
4168:
4166:
4163:
4160:
4159:
4146:
4137:
4126:
4125:
4123:
4120:
4118:
4115:
4111:Koch snowflake
4099:
4096:
4072:
4052:
4049:
4046:
4043:
4040:
4037:
4034:
4029:
4025:
3997:saddle surface
3966:
3965:
3956:
3955:
3947:
3946:
3945:
3944:
3943:
3931:
3928:
3911:
3908:
3905:
3902:
3882:
3879:
3876:
3873:
3853:
3830:pseudotriangle
3791:Main article:
3788:
3785:
3783:
3780:
3776:convex polygon
3767:is the unique
3734:
3733:
3724:
3723:
3715:
3714:
3713:
3712:
3711:
3709:
3706:
3689:
3686:
3666:
3655:convex polygon
3640:cyclic hexagon
3612:
3608:
3602:
3597:
3577:
3557:
3537:
3533:
3529:
3526:
3523:
3503:
3500:
3497:
3492:
3488:
3467:
3459:
3455:
3451:
3448:
3441:
3437:
3433:
3427:
3421:
3418:
3415:
3410:
3406:
3400:
3397:
3394:
3388:
3383:
3379:
3358:
3338:
3329:from the side
3316:
3312:
3291:
3269:
3265:
3244:
3222:
3218:
3181:pedal triangle
3168:pedal triangle
3165:
3164:
3155:
3154:
3146:
3145:
3144:
3143:
3142:
3130:
3127:
3119:
3115:
3112:
3106:
3101:
3097:
3094:
3084:
3080:
3077:
3071:
3066:
3062:
3059:
3050:
3042:
3038:
3035:
3029:
3024:
3020:
3017:
3007:
3003:
3000:
2994:
2989:
2985:
2982:
2973:
2965:
2961:
2958:
2952:
2947:
2943:
2940:
2930:
2926:
2923:
2917:
2912:
2908:
2905:
2879:
2859:
2839:
2836:
2833:
2805:
2802:
2800:
2797:
2796:
2795:
2783:
2780:
2777:
2774:
2771:
2757:
2745:
2742:
2739:
2719:
2716:
2713:
2710:
2707:
2680:
2677:
2656:
2653:
2650:
2630:
2627:
2624:
2603:
2591:simple polygon
2568:Main article:
2565:
2562:
2518:Main article:
2515:
2512:
2499:Main article:
2496:
2493:
2475:
2471:
2467:
2440:
2437:
2432:
2428:
2422:
2418:
2414:
2409:
2405:
2399:
2395:
2391:
2386:
2382:
2376:
2372:
2368:
2363:
2359:
2353:
2349:
2345:
2340:
2336:
2330:
2326:
2322:
2317:
2313:
2307:
2303:
2299:
2293:
2290:
2284:
2281:
2279:
2277:
2272:
2264:
2260:
2256:
2252:
2248:
2244:
2243:
2238:
2234:
2230:
2226:
2222:
2218:
2217:
2215:
2207:
2204:
2198:
2193:
2185:
2181:
2177:
2173:
2169:
2165:
2164:
2159:
2155:
2151:
2147:
2143:
2139:
2138:
2136:
2128:
2125:
2119:
2114:
2106:
2102:
2098:
2094:
2090:
2086:
2085:
2080:
2076:
2072:
2068:
2064:
2060:
2059:
2057:
2049:
2046:
2040:
2035:
2029:
2026:
2024:
2021:
2019:
2016:
2015:
2010:
2006:
2002:
1998:
1994:
1990:
1986:
1982:
1978:
1977:
1972:
1968:
1964:
1960:
1956:
1952:
1948:
1944:
1940:
1939:
1937:
1929:
1926:
1920:
1917:
1915:
1913:
1910:
1909:
1881:
1876:
1872:
1868:
1863:
1859:
1855:
1831:
1826:
1822:
1818:
1813:
1809:
1805:
1781:
1776:
1772:
1768:
1763:
1759:
1755:
1673:
1668:
1665:
1662:
1659:
1656:
1653:
1650:
1647:
1644:
1641:
1638:
1635:
1632:
1629:
1626:
1623:
1618:
1615:
1591:
1588:
1585:
1582:
1579:
1576:
1573:
1567:
1564:
1558:
1555:
1535:
1515:
1495:
1482:, named after
1466:
1463:
1460:
1457:
1454:
1451:
1445:
1442:
1436:
1433:
1411:
1408:
1405:
1402:
1399:
1396:
1393:
1390:
1368:
1346:
1322:
1282:
1258:
1234:
1231:
1228:
1222:
1219:
1213:
1210:
1188:
1164:
1140:
1111:Main article:
1108:
1105:
1104:
1103:
1096:
1093:
1086:
1067:
1066:
1063:
1060:included angle
1055:
1052:If and only if
1018:
1015:
1003:
1000:
997:
994:
991:
988:
985:
982:
979:
976:
973:
970:
967:
964:
961:
958:
955:
952:
949:
946:
943:
938:
934:
930:
927:
924:
919:
915:
911:
908:
905:
900:
896:
885:if and only if
872:
852:
832:
812:
792:
772:
750:law of cosines
718:exterior angle
692:
689:
673:center of mass
639:
638:
629:
628:
620:
619:
618:
617:
616:
559:angle bisector
542:
541:
532:
531:
523:
522:
521:
520:
519:
430:Ceva's theorem
421:Main article:
418:
415:
413:
410:
409:
408:
403:
396:
394:
391:Acute triangle
389:
382:
380:
377:Right triangle
375:
368:
365:
364:
361:
354:
352:
347:
340:
338:
333:
326:
312:acute triangle
308:right triangle
277:
274:
144:straight angle
91:
90:
83:
77:
76:
73:
67:
66:
63:
53:
52:
44:
43:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
6996:
6985:
6982:
6981:
6979:
6964:
6963:Weakly simple
6961:
6959:
6956:
6954:
6951:
6949:
6946:
6944:
6941:
6939:
6936:
6934:
6931:
6929:
6926:
6924:
6921:
6919:
6916:
6914:
6911:
6909:
6906:
6904:
6903:Infinite skew
6901:
6899:
6896:
6894:
6891:
6889:
6886:
6884:
6881:
6879:
6876:
6875:
6873:
6869:
6863:
6860:
6858:
6855:
6853:
6850:
6848:
6845:
6843:
6840:
6838:
6835:
6833:
6830:
6828:
6825:
6824:
6822:
6819:
6818:Star polygons
6815:
6805:
6804:Apeirogon (∞)
6802:
6800:
6797:
6795:
6792:
6790:
6787:
6785:
6782:
6780:
6777:
6775:
6772:
6770:
6767:
6765:
6762:
6761:
6759:
6755:
6749:
6748:Icosagon (20)
6746:
6744:
6741:
6739:
6736:
6734:
6731:
6729:
6726:
6724:
6721:
6719:
6716:
6714:
6711:
6709:
6706:
6705:
6703:
6699:
6693:
6690:
6688:
6685:
6683:
6680:
6678:
6675:
6673:
6670:
6668:
6665:
6663:
6660:
6658:
6655:
6653:
6650:
6648:
6645:
6644:
6642:
6638:
6635:
6629:
6623:
6620:
6618:
6615:
6613:
6610:
6608:
6605:
6603:
6600:
6598:
6595:
6593:
6590:
6588:
6585:
6583:
6582:Parallelogram
6580:
6578:
6577:Orthodiagonal
6575:
6573:
6570:
6568:
6565:
6563:
6560:
6558:
6557:Ex-tangential
6555:
6553:
6550:
6548:
6545:
6543:
6540:
6538:
6535:
6533:
6530:
6529:
6527:
6525:
6521:
6515:
6512:
6510:
6507:
6505:
6502:
6500:
6497:
6495:
6492:
6490:
6487:
6485:
6482:
6481:
6479:
6477:
6473:
6468:
6464:
6457:
6452:
6450:
6445:
6443:
6438:
6437:
6434:
6427:
6423:
6419:
6415:
6414:
6409:
6404:
6403:
6399:
6392:
6386:
6382:
6381:
6376:
6372:
6368:
6363:
6359:
6353:
6349:
6345:
6341:
6337:
6332:
6328:
6326:9781607526001
6322:
6318:
6314:
6310:
6306:
6304:0-13-032927-4
6300:
6296:
6291:
6287:
6285:0-471-25183-6
6281:
6277:
6272:
6268:
6262:
6258:
6254:
6253:
6247:
6243:
6237:
6233:
6229:
6225:
6220:
6216:
6210:
6206:
6205:
6198:
6194:
6190:
6189:
6183:
6179:
6175:
6171:
6167:
6163:
6159:
6158:
6152:
6148:
6142:
6138:
6134:
6130:
6126:
6121:
6117:
6113:
6109:
6105:
6101:
6097:
6093:
6089:
6088:
6082:
6078:
6074:
6070:
6066:
6065:
6059:
6055:
6049:
6045:
6041:
6037:
6033:
6029:
6025:
6023:9781470464431
6019:
6015:
6011:
6010:
6004:
6000:
5996:
5992:
5988:
5987:
5982:
5977:
5973:
5967:
5963:
5959:
5958:
5952:
5948:
5944:
5940:
5936:
5932:
5928:
5923:
5918:
5914:
5910:
5909:
5903:
5899:
5893:
5889:
5885:
5881:
5877:
5876:
5870:
5867:
5863:
5859:
5854:
5850:
5846:
5845:
5840:
5836:
5832:
5826:
5822:
5821:
5815:
5811:
5805:
5801:
5797:
5796:Gonick, Larry
5793:
5789:
5783:
5779:
5775:
5774:
5768:
5764:
5758:
5754:
5750:
5749:
5743:
5739:
5733:
5729:
5725:
5721:
5717:
5713:
5709:
5705:
5701:
5697:
5693:
5689:
5684:
5680:
5674:
5670:
5669:
5664:
5663:Byrne, Oliver
5660:
5656:
5650:
5646:
5641:
5637:
5631:
5627:
5622:
5618:
5612:
5608:
5604:
5600:
5595:
5591:
5587:
5583:
5581:3-7643-5242-6
5577:
5573:
5569:
5565:
5561:
5557:
5553:
5549:
5548:
5542:
5538:
5532:
5528:
5524:
5523:
5517:
5513:
5511:0-471-17421-1
5507:
5503:
5499:
5495:
5491:
5485:
5481:
5477:
5473:
5468:
5464:
5460:
5456:
5452:
5451:
5445:
5444:
5439:
5431:
5425:
5421:
5420:
5412:
5409:
5405:
5404:Ballmann 1995
5400:
5397:
5393:
5388:
5385:
5373:
5366:
5363:
5351:
5347:
5340:
5337:
5333:
5329:
5324:
5322:
5318:
5314:
5310:
5305:
5302:
5298:
5294:
5289:
5286:
5282:
5278:
5273:
5270:
5264:
5261:
5259:
5255:
5252:
5251:
5247:
5244:
5240:
5235:
5232:
5227:
5223:
5219:
5215:
5208:
5205:
5200:
5196:
5192:
5188:
5181:
5178:
5173:
5169:
5165:
5161:
5157:
5153:
5146:
5143:
5139:
5134:
5131:
5127:
5122:
5119:
5113:
5110:
5108:
5105:
5104:
5100:
5097:
5092:
5088:
5081:
5074:
5071:
5066:
5062:
5058:
5054:
5050:
5046:
5042:
5035:
5032:
5028:
5023:
5020:
5016:
5011:
5008:
5004:
4999:
4996:
4990:
4986:
4983:
4981:
4978:
4977:
4973:
4970:
4966:
4961:
4958:
4954:
4953:Meisters 1975
4949:
4946:
4942:
4937:
4934:
4930:
4925:
4922:
4918:
4914:
4909:
4906:
4902:
4897:
4894:
4887:
4884:
4881:
4878:
4877:
4873:
4870:
4866:
4861:
4858:
4854:
4850:
4849:
4844:
4840:
4833:
4830:
4826:
4822:
4817:
4814:
4810:
4805:
4802:
4799:, p. 64.
4798:
4793:
4790:
4786:
4781:
4778:
4774:
4769:
4766:
4762:
4757:
4754:
4748:
4745:
4743:
4740:
4739:
4735:
4732:
4728:
4724:
4719:
4716:
4712:
4708:
4703:
4700:
4696:
4691:
4688:
4684:
4680:
4675:
4672:
4668:
4663:
4660:
4656:
4651:
4648:
4644:
4640:
4635:
4632:
4628:
4624:
4619:
4616:
4612:
4608:
4604:
4600:
4596:
4592:
4588:
4587:
4579:
4576:
4571:
4565:
4561:
4560:
4552:
4549:
4545:
4541:
4536:
4534:
4530:
4526:
4522:
4517:
4514:
4510:
4506:
4501:
4498:
4494:
4490:
4485:
4482:
4476:
4472:
4469:
4467:
4463:
4460:
4459:
4455:
4452:
4448:
4444:
4439:
4436:
4432:
4428:
4423:
4420:
4416:
4412:
4407:
4404:
4400:
4399:126–127
4396:
4391:
4388:
4384:
4380:
4375:
4372:
4368:
4364:
4359:
4356:
4352:
4347:
4344:
4338:
4334:
4331:
4329:
4325:
4322:
4321:
4317:
4314:
4307:
4304:
4302:
4298:
4295:
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3751:
3750:tangent lines
3747:
3738:
3728:
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3702:parallelogram
3687:
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3159:
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3141:
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3095:
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2674:
2670:
2654:
2651:
2648:
2628:
2625:
2622:
2601:
2592:
2588:
2584:
2583:Triangulation
2576:
2571:
2563:
2561:
2559:
2555:
2554:cantilevering
2551:
2547:
2543:
2538:
2535:
2534:parallelogram
2526:
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2513:
2511:
2508:
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2250:
2246:
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2196:
2191:
2183:
2179:
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2100:
2092:
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2078:
2074:
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2062:
2055:
2047:
2044:
2038:
2033:
2027:
2022:
2017:
2008:
2004:
1996:
1992:
1984:
1980:
1970:
1966:
1958:
1954:
1946:
1942:
1935:
1927:
1924:
1918:
1916:
1911:
1899:
1897:
1874:
1870:
1866:
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1857:
1824:
1820:
1816:
1811:
1807:
1774:
1770:
1766:
1761:
1757:
1743:
1739:
1734:
1732:
1728:
1727:parallelogram
1724:
1723:oriented area
1720:
1716:
1708:
1697:share a base
1695:
1688:
1684:
1671:
1663:
1660:
1657:
1648:
1645:
1642:
1633:
1630:
1627:
1621:
1616:
1613:
1605:
1604:semiperimeter
1586:
1583:
1580:
1577:
1574:
1565:
1562:
1556:
1553:
1533:
1513:
1493:
1485:
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1477:
1464:
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1455:
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1440:
1434:
1431:
1406:
1400:
1397:
1394:
1391:
1388:
1366:
1344:
1320:
1311:If two sides
1306:
1300:
1296:
1280:
1256:
1245:
1232:
1229:
1226:
1220:
1217:
1211:
1208:
1186:
1162:
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1128:
1119:
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1106:
1101:
1097:
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1087:
1084:
1083:
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1080:
1075:
1072:
1064:
1061:
1056:
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1049:
1048:
1046:
1041:
1039:
1038:
1029:
1023:
1016:
1014:
1001:
998:
992:
986:
983:
977:
971:
968:
962:
956:
953:
950:
947:
944:
941:
936:
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922:
917:
913:
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906:
903:
898:
894:
886:
870:
850:
830:
810:
790:
770:
762:
758:
753:
751:
747:
743:
739:
735:
730:
728:
724:
723:supplementary
720:
719:
714:
710:
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697:
690:
688:
686:
685:
680:
679:
674:
670:
669:
664:
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659:
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607:
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583:
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549:
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462:
458:
454:
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431:
424:
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411:
406:
400:
395:
392:
386:
381:
378:
372:
367:
358:
353:
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344:
339:
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330:
325:
323:
321:
317:
313:
309:
305:
301:
297:
293:
288:
286:
285:
275:
273:
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267:
263:
258:
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235:
234:
229:
225:
221:
217:
213:
209:
204:
202:
198:
194:
190:
186:
185:straight line
182:
177:
175:
171:
170:
165:
164:
159:
158:
153:
152:planar region
149:
145:
141:
137:
133:
132:line segments
129:
128:
123:
120:
116:
115:
110:
106:
102:
98:
89:
84:
82:
78:
74:
72:
68:
64:
62:
58:
54:
50:
45:
40:
37:
33:
19:
6757:>20 sides
6692:Decagon (10)
6677:Heptagon (7)
6667:Pentagon (5)
6657:Triangle (3)
6656:
6552:Equidiagonal
6475:
6411:
6380:Trigonometry
6379:
6366:
6339:
6316:
6294:
6275:
6251:
6226:. Springer.
6223:
6202:
6192:
6186:
6161:
6155:
6128:
6091:
6085:
6068:
6062:
6035:
6008:
5990:
5984:
5956:
5912:
5906:
5874:
5843:
5819:
5799:
5772:
5747:
5723:
5691:
5687:
5667:
5644:
5625:
5601:. Springer.
5598:
5571:
5551:
5545:
5521:
5501:
5474:. Springer.
5471:
5454:
5448:
5418:
5411:
5399:
5387:
5375:. Retrieved
5370:Wood, John.
5365:
5353:. Retrieved
5349:
5339:
5328:Nielsen 2021
5304:
5288:
5272:
5246:
5234:
5217:
5213:
5207:
5190:
5186:
5180:
5155:
5151:
5145:
5133:
5121:
5099:
5090:
5086:
5073:
5048:
5044:
5034:
5022:
5010:
4998:
4985:Ericson 2005
4980:Oldknow 1995
4972:
4965:Oldknow 1995
4960:
4948:
4936:
4924:
4908:
4896:
4886:Apostol 1997
4882:, p. 80
4872:
4860:
4846:
4832:
4816:
4804:
4792:
4780:
4768:
4756:
4734:
4718:
4702:
4690:
4674:
4662:
4650:
4634:
4618:
4610:
4590:
4584:
4578:
4558:
4551:
4516:
4500:
4484:
4454:
4438:
4422:
4406:
4390:
4374:
4358:
4346:
4316:
4289:
4273:
4257:
4227:
4200:
4188:
4172:
4149:
4140:
4130:
4101:
4092:CAT(k) space
4085:
4009:
3978:
3837:
3827:
3804:
3773:
3762:
3754:
3743:
3652:
3633:
3568:is equal to
3204:
3201:
3190:
3178:
2807:
2762:of the form
2689:
2682:
2581:
2542:tessellating
2539:
2531:
2504:
2456:
1900:
1735:
1730:
1719:affine plane
1712:
1693:
1478:
1310:
1304:
1246:
1124:
1099:
1076:
1068:
1059:
1042:
1035:
1033:
754:
746:law of sines
731:
716:
702:
682:
676:
666:
656:
650:
648:
611:Euler's line
609:
595:
589:
579:
575:
569:
563:
557:
555:
548:Euler's line
512:
508:
504:
498:
496:
486:circumcircle
484:
480:circumcenter
478:
468:
466:
453:circumcenter
426:
299:
289:
282:
279:
262:trigonometry
259:
255:circular-arc
237:
231:
205:
199:determine a
178:
167:
161:
155:
148:plane figure
125:
112:
96:
94:
36:
6953:Star-shaped
6928:Rectilinear
6898:Equilateral
6893:Equiangular
6857:Hendecagram
6701:11–20 sides
6682:Octagon (8)
6672:Hexagon (6)
6647:Monogon (1)
6489:Equilateral
6071:: 119–120.
6032:Lang, Serge
5866:486-60088-2
5457:: 161–165.
5440:Works cited
5392:Berger 2002
5093:(1): 15–24.
5003:Kalman 2008
4989:46–47
4929:Gonick 2024
4880:Gonick 2024
4809:Gonick 2024
4797:Gonick 2024
4785:Gonick 2024
4773:Gonick 2024
4761:Gonick 2024
4695:Gonick 2024
4667:Gonick 2024
4308:, p. 4
3924:convex hull
3838:cusp points
1271:and height
1043:Some basic
1028:hatch marks
597:orthocenter
514:orthocenter
457:orthocenter
304:right angle
240:curvilinear
201:tetrahedron
119:dimensional
117:, are zero-
6958:Tangential
6862:Dodecagram
6640:1–10 sides
6631:By number
6612:Tangential
6592:Right kite
6408:"Triangle"
6195:: 113–115.
5330:, p.
5311:, p.
5295:, p.
5279:, p.
5256:, p.
4987:, p.
4915:, p.
4901:Smith 2000
4865:Heath 1926
4823:, p.
4725:, p.
4723:Axler 2012
4709:, p.
4707:Young 2017
4681:, p.
4655:Heath 1926
4641:, p.
4639:Holme 2010
4625:, p.
4542:, p.
4523:, p.
4507:, p.
4491:, p.
4473:, p.
4464:, p.
4445:, p.
4429:, p.
4413:, p.
4397:, p.
4381:, p.
4379:Holme 2010
4365:, p.
4363:Holme 2010
4351:Heath 1926
4335:, p.
4326:, p.
4299:, p.
4278:Stahl 2003
4262:Heath 1926
4246:, p.
4237:, p.
4217:Heath 1926
4210:, p.
4193:Byrne 2013
4179:, p.
4117:References
3657:with area
2495:Inequality
1546:. Letting
434:concurrent
412:Properties
18:Triangular
6984:Triangles
6938:Reinhardt
6847:Enneagram
6837:Heptagram
6827:Pentagram
6794:65537-gon
6652:Digon (2)
6622:Trapezoid
6587:Rectangle
6537:Bicentric
6499:Isosceles
6476:Triangles
6418:EMS Press
6278:. Wiley.
5917:CiteSeerX
5778:CRC Press
5665:(2013) .
5504:. Wiley.
5377:19 August
5355:19 August
5254:Hann 2014
5065:0025-5572
4821:Ryan 2008
4623:Ryan 2008
4540:King 2021
4521:Ryan 2008
4505:Ryan 2008
4489:King 2021
4471:King 2021
4443:Ryan 2008
4333:Ryan 2008
4297:Ryan 2008
4244:Ryan 2008
4165:Footnotes
4135:triangle.
4033:×
4028:∘
3907:−
3878:−
3118:¯
3105:⋅
3100:¯
3083:¯
3070:⋅
3065:¯
3041:¯
3028:⋅
3023:¯
3006:¯
2993:⋅
2988:¯
2964:¯
2951:⋅
2946:¯
2929:¯
2916:⋅
2911:¯
2782:γ
2776:β
2770:α
2652:−
2626:−
2470:⋅
2413:−
2367:−
2321:−
1740:(such as
1661:−
1646:−
1631:−
1462:γ
1459:
1407:γ
1401:
1367:γ
1071:congruent
993:γ
987:
978:β
972:
963:α
957:
945:γ
942:
926:β
923:
907:α
904:
871:γ
851:β
831:α
811:γ
791:β
771:α
761:collinear
678:symmedian
606:excircles
581:excircles
438:collinear
233:geodesics
212:curvature
88:see below
6978:Category
6913:Isotoxal
6908:Isogonal
6852:Decagram
6842:Octagram
6832:Hexagram
6633:of sides
6562:Harmonic
6463:Polygons
6377:(2017).
5880:Springer
5841:(1926).
5798:(2024).
5722:(1967).
5570:(1995).
5500:(1997).
5172:40378417
4109:and the
4063:, where
3205:distinct
2890:, then:
2546:hexagons
2514:Rigidity
1731:Elements
1602:be the
1045:theorems
748:and the
668:centroid
663:midpoint
661:and the
642:centroid
576:inradius
571:incircle
565:incenter
500:altitude
475:midpoint
461:incircle
284:Elements
243:triangle
114:vertices
109:geometry
97:triangle
61:vertices
42:Triangle
6933:Regular
6878:Concave
6871:Classes
6779:257-gon
6602:Rhombus
6542:Crossed
6204:College
6178:3618298
6116:0367792
6108:2319703
5947:1299166
5939:2974536
5708:1168956
5590:1377265
4607:2430410
4103:Fractal
2487:is the
1892:
1846:
1842:
1796:
1792:
1746:
1422:
1381:
1357:
1337:
1333:
1313:
1293:
1273:
1269:
1249:
1199:
1179:
1175:
1155:
1151:
1131:
1100:AAcorrS
1037:similar
883:exists
440:; here
228:surface
101:polygon
6943:Simple
6888:Cyclic
6883:Convex
6607:Square
6547:Cyclic
6509:Obtuse
6504:Kepler
6387:
6354:
6323:
6301:
6282:
6263:
6238:
6211:
6176:
6143:
6114:
6106:
6050:
6020:
5968:
5945:
5937:
5919:
5894:
5864:
5827:
5806:
5784:
5759:
5734:
5706:
5675:
5651:
5632:
5613:
5588:
5578:
5533:
5508:
5486:
5426:
5170:
5063:
4605:
4566:
4155:convex
4005:sphere
3653:Every
2756:, etc.
2550:nature
2457:where
1736:Given
863:, and
803:, and
691:Angles
658:vertex
652:median
320:Euclid
247:curved
172:. The
169:height
122:points
105:shapes
6918:Magic
6514:Right
6494:Ideal
6484:Acute
6174:JSTOR
6104:JSTOR
5935:JSTOR
5168:JSTOR
5083:(PDF)
4282:p. 37
4122:Notes
3815:disks
3769:conic
3638:is a
2593:with
306:is a
253:with
236:). A
193:plane
127:edges
99:is a
57:Edges
6948:Skew
6572:Kite
6467:List
6385:ISBN
6369:(6).
6352:ISBN
6321:ISBN
6299:ISBN
6280:ISBN
6261:ISBN
6236:ISBN
6209:ISBN
6141:ISBN
6048:ISBN
6018:ISBN
5966:ISBN
5892:ISBN
5825:ISBN
5804:ISBN
5782:ISBN
5757:ISBN
5732:ISBN
5673:ISBN
5649:ISBN
5630:ISBN
5611:ISBN
5576:ISBN
5531:ISBN
5506:ISBN
5484:ISBN
5424:ISBN
5379:2024
5357:2024
5061:ISSN
4564:ISBN
3987:and
3970:and
3938:and
3763:The
3744:The
3634:The
3191:The
3170:and
3166:The
2870:and
2505:The
1335:and
1127:area
1107:Area
703:The
509:foot
505:base
451:The
222:. A
163:apex
157:base
81:Area
59:and
6344:doi
6228:doi
6166:doi
6133:doi
6096:doi
6073:doi
6040:doi
5995:doi
5991:115
5927:doi
5913:101
5884:doi
5862:SBN
5849:hdl
5696:doi
5603:doi
5556:doi
5476:doi
5459:doi
5332:154
5297:259
5222:doi
5195:doi
5191:101
5160:doi
5053:doi
4917:834
4727:634
4643:240
4627:102
4595:doi
4544:155
4525:104
4509:106
4493:153
4447:105
4415:128
4383:143
4367:210
4024:180
3811:arc
2669:ear
1694:ABC
1456:sin
1398:sin
984:cos
969:cos
954:cos
933:cos
914:cos
895:cos
707:in
556:An
497:An
218:or
206:In
203:.
179:In
107:in
6980::
6416:.
6410:.
6350:.
6342:.
6259:.
6255:.
6234:.
6193:13
6191:.
6172:.
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