7633:, which is to calculate the length of a belt or rope needed to fit snugly over two pulleys. If the belt is considered to be a mathematical line of negligible thickness, and if both pulleys are assumed to lie in exactly the same plane, the problem devolves to summing the lengths of the relevant tangent line segments with the lengths of circular arcs subtended by the belt. If the belt is wrapped about the wheels so as to cross, the interior tangent line segments are relevant. Conversely, if the belt is wrapped exteriorly around the pulleys, the exterior tangent line segments are relevant; this case is sometimes called the
4283:, whereas the internal tangent lines intersect at the internal homothetic center. Both the external and internal homothetic centers lie on the line of centers (the line connecting the centers of the two circles), closer to the center of the smaller circle: the internal center is in the segment between the two circles, while the external center is not between the points, but rather outside, on the side of the center of the smaller circle. If the two circles have equal radius, there are still four bitangents, but the external tangent lines are parallel and there is no external center in the
4256:
4304:
235:
160:
5170:
4895:
2197:
1536:
1479:
5292:
7788:
5050:
3800:
7153:
6747:
4052:
4885:
1041:
4628:
1018:
6813:
3222:
7746:
7599:
but considered "inside out"), in which case if the radii have opposite sign (one circle has negative radius and the other has positive radius) the external and internal homothetic centers and external and internal bitangents are switched, while if the radii have the same sign (both positive radii or
7844:
A third generalization considers tangent circles, rather than tangent lines; a tangent line can be considered as a tangent circle of infinite radius. In particular, the external tangent lines to two circles are limiting cases of a family of circles which are internally or externally tangent to both
7211:
Two distinct circles may have between zero and four bitangent lines, depending on configuration; these can be classified in terms of the distance between the centers and the radii. If counted with multiplicity (counting a common tangent twice) there are zero, two, or four bitangent lines. Bitangent
6416:
Geometrically this corresponds to computing the angle formed by the tangent lines and the line of centers, and then using that to rotate the equation for the line of centers to yield an equation for the tangent line. The angle is computed by computing the trigonometric functions of a right triangle
4278:
there may be any number between zero and four bitangent lines; these are addressed below. For two of these, the external tangent lines, the circles fall on the same side of the line; for the two others, the internal tangent lines, the circles fall on opposite sides of the line. The external tangent
3854:
7749:
Animation showing the inversive transformation of an
Apollonius problem. The blue and red circles swell to tangency, and are inverted in the grey circle, producing two straight lines. The yellow solutions are found by sliding a circle between them until it touches the transformed green circle from
7607:
Note that in these degenerate cases the external and internal homothetic center do generally still exist (the external center is at infinity if the radii are equal), except if the circles coincide, in which case the external center is not defined, or if both circles have radius zero, in which case
5065:
The bitangent lines can be constructed either by constructing the homothetic centers, as described at that article, and then constructing the tangent lines through the homothetic center that is tangent to one circle, by one of the methods described above. The resulting line will then be tangent to
7603:
Bitangent lines can also be defined when one or both of the circles has radius zero. In this case the circle with radius zero is a double point, and thus any line passing through it intersects the point with multiplicity two, hence is "tangent". If one circle has radius zero, a bitangent line is
4661:
7762:
problem). To solve this problem, the center of any such circle must lie on an angle bisector of any pair of the lines; there are two angle-bisecting lines for every intersection of two lines. The intersections of these angle bisectors give the centers of solution circles. There are four such
7778:
in their tangent point with respect to a circle of appropriate radius transforms the two touching given circles into two parallel lines, and the third given circle into another circle. Thus, the solutions may be found by sliding a circle of constant radius between two parallel lines until it
4229:
7324:) – have one point of external tangency – then they have two external bitangents and one internal bitangent, namely the common tangent line. This common tangent line has multiplicity two, as it separates the circles (one on the left, one on the right) for either orientation (direction).
2653:
6414:
2478:
2975:
4428:
151:
The radius of a circle is perpendicular to the tangent line through its endpoint on the circle's circumference. Conversely, the perpendicular to a radius through the same endpoint is a tangent line. The resulting geometrical figure of circle and tangent line has a
782:
1474:{\displaystyle {\begin{aligned}y&=-{\frac {x_{1}-a}{y_{1}-b}}x+y_{1}+x_{1}{\frac {x_{1}-a}{y_{1}-b}}\\y-y_{1}&=(x_{1}-x){\frac {x_{1}-a}{y_{1}-b}}\\(y-y_{1})(y_{1}-b)&=-(x-x_{1})(x_{1}-a)\\(x-x_{1})(x_{1}-a)+(y-y_{1})(y_{1}-b)&=0\end{aligned}}}
2989:
6625:
5697:
6133:
4059:
7604:
simply a line tangent to the circle and passing through the point, and is counted with multiplicity two. If both circles have radius zero, then the bitangent line is the line they define, and is counted with multiplicity four.
3795:
2483:
7148:{\displaystyle {\begin{aligned}(t_{2}-v_{2})\cdot (t_{2}-t_{1})&=0\\(t_{1}-v_{1})\cdot (t_{2}-t_{1})&=0\\(t_{1}-v_{1})\cdot (t_{1}-v_{1})&=r_{1}^{2}\\(t_{2}-v_{2})\cdot (t_{2}-v_{2})&=r_{2}^{2}\\\end{aligned}}}
6226:
6417:
whose vertices are the (external) homothetic center, a center of a circle, and a tangent point; the hypotenuse lies on the tangent line, the radius is opposite the angle, and the adjacent side lies on the line of centers.
6238:
7220:
can also be understood in terms of limits of other configurations – e.g., a limit of two circles that almost touch, and moving one so that they touch, or a circle with small radius shrinking to a circle of zero radius.
2331:
4047:{\displaystyle {\begin{aligned}{\overline {BP}}={\overline {BQ}}=b,&\quad {\overline {CQ}}={\overline {CR}}=c,\\{\overline {DR}}={\overline {DS}}=d,&\quad {\overline {AS}}={\overline {AP}}=a.\end{aligned}}}
1046:
4880:{\displaystyle {\begin{aligned}\gamma &=-{\text{atan2}}\left({y_{2}-y_{1}},{x_{2}-x_{1}}\right)\\\beta &=\pm \arcsin \left({\frac {R-r}{\sqrt {(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}}}\right)\end{aligned}}}
787:
5858:
4666:
3450:
2799:
5773:
573:
6027:
7990:
6818:
6243:
6143:
5574:
4433:
4064:
3859:
7527:) then they have no bitangents, as a tangent line to the outer circle does not intersect the inner circle, or conversely a tangent line to the inner circle is a secant line to the outer circle.
771:
7407:
5070:
these constructions break down; to simplify exposition this is not discussed in this section, but a form of the construction can work in limit cases (e.g., two circles tangent at one point).
2794:
6537:
5569:
7825:
The concept of a tangent line to one or more circles can be generalized in several ways. First, the conjugate relationship between tangent points and tangent lines can be generalized to
5396:
at the desired tangent points. The desired internal tangent lines are the lines perpendicular to these radial lines at those tangent points, which may be constructed as described above.
5283:
at the desired tangent points. The desired external tangent lines are the lines perpendicular to these radial lines at those tangent points, which may be constructed as described above.
4623:{\displaystyle {\begin{aligned}x_{3}&=x_{1}\pm r\sin \alpha \\y_{3}&=y_{1}\pm r\cos \alpha \\x_{4}&=x_{2}\pm R\sin \alpha \\y_{4}&=y_{2}\pm R\cos \alpha \\\end{aligned}}}
6050:
659:
4980:
427:
8129:
7860:
which maps one to the other. In Möbius geometry, tangency between a line and a circle becomes a special case of tangency between two circles. This equivalence is extended further in
7597:
6509:
2090:
1013:{\displaystyle {\begin{aligned}{\overset {}{(}}x-a)^{2}+(y-b)^{2}&=r^{2}\\2(x-a)+2(y-b){\frac {dy}{dx}}&=0\\{\frac {dy}{dx}}&=-{\frac {x_{1}-a}{y_{1}-b}}\end{aligned}}}
5956:
5907:
3325:
2326:
2714:
7525:
7468:) – have one point of internal tangency – then they have no internal bitangents and one external bitangent, namely the common tangent line, which has multiplicity two, as above.
1975:
8046:
7845:
circles, while the internal tangent lines are limiting cases of a family of circles which are internally tangent to one and externally tangent to the other of the two circles.
5057:
An inner tangent is a tangent that intersects the segment joining two circles' centers. Note that the inner tangent will not be defined for cases when the two circles overlap.
7269:
2257:
7466:
2146:
5564:
7322:
3638:
2192:
7198:
3502:
2027:
3711:
5515:
3544:
3217:{\displaystyle {\binom {x_{1/2}}{y_{1/2}}}={\frac {r^{2}}{d_{0}^{2}}}{\binom {x_{0}}{y_{0}}}\pm {\frac {r}{d_{0}^{2}}}{\sqrt {d_{0}^{2}-r^{2}}}{\binom {-y_{0}}{x_{0}}}.}
3592:
6138:
1919:
7531:
Finally, if the two circles are identical, any tangent to the circle is a common tangent and hence (external) bitangent, so there is a circle's worth of bitangents.
7766:
A general
Apollonius problem can be transformed into the simpler problem of circle tangent to one circle and two parallel lines (itself a special case of the
120:
intersect a circle at two points, whereas another line may not intersect a circle at all. This property of tangent lines is preserved under many geometrical
7758:
involve finding a circle that is tangent to one or more lines. The simplest of these is to construct circles that are tangent to three given lines (the
5220:
that are tangent to this new circle. These lines are parallel to the desired tangent lines, because the situation corresponds to shrinking both circles
4224:{\displaystyle {\begin{aligned}&{\overline {AB}}+{\overline {CD}}=(a+b)+(c+d)\\={}&{\overline {BC}}+{\overline {DA}}=(b+c)+(d+a)\end{aligned}}}
7409:), then they have no internal bitangents and two external bitangents (they cannot be separated, because they intersect, hence no internal bitangents).
5778:
3807:
This conclusion follows from the equality of the tangent segments from the four vertices of the quadrilateral. Let the tangent points be denoted as
7600:
both negative radii) "external" and "internal" have the same usual sense (switching one sign switches them, so switching both switches them back).
4239:
This theorem and its converse have various uses. For example, they show immediately that no rectangle can have an inscribed circle unless it is a
199:
outside of the circle. The geometrical figure of a circle and both tangent lines likewise has a reflection symmetry about the radial axis joining
7740:
5066:
the other circle as well. Alternatively, the tangent lines and tangent points can be constructed more directly, as detailed below. Note that in
1697:
1551:
1489:
55:
5718:
2648:{\displaystyle {\binom {a}{b_{\pm }}}={\frac {r^{2}}{x_{0}}}{\binom {1}{0}}\pm {\frac {r}{x_{0}}}{\sqrt {x_{0}^{2}-r^{2}}}{\binom {0}{1}}\ .}
464:
5342:
that are tangent to this new circle. These lines are parallel to the desired tangent lines, because the situation corresponds to shrinking
4236:
The converse is also true: a circle can be inscribed into every quadrilateral in which the lengths of opposite sides sum to the same value.
1022:
Now that we have the slope of the tangent line, we can substitute the slope and the coordinate of the tangency point into the line equation
7763:
circles in general, the inscribed circle of the triangle formed by the intersection of the three lines, and the three exscribed circles.
1763:
be the six intersection points, with the same letter corresponding to the same line and the index 1 corresponding to the point closer to
7829:, in which the pole points may be anywhere, not only on the circumference of the circle. Second, the union of two circles is a special (
6721:
to the right, and the two tangent lines will cross. In this way all four solutions are obtained. Switching signs of both radii switches
6409:{\displaystyle {\begin{aligned}a&=RX-kY{\sqrt {1-R^{2}}}\\b&=RY+kX{\sqrt {1-R^{2}}}\\c&=r_{1}-(ax_{1}+by_{1})\end{aligned}}}
7203:
These are four quadratic equations in two two-dimensional vector variables, and in general position will have four pairs of solutions.
5161:
is defined as the larger of the two circles. Two different methods may be used to construct the external and internal tangent lines.
2473:{\displaystyle a={\frac {r^{2}}{x_{0}}},\qquad b_{\pm }=\pm {\sqrt {r^{2}-a^{2}}}=\pm {\frac {r}{x_{0}}}{\sqrt {x_{0}^{2}-r^{2}}}.}
2970:{\textstyle {\vec {e}}_{1}={\frac {1}{d_{0}}}{\binom {x_{0}}{y_{0}}},\ {\vec {e}}_{2}={\frac {1}{d_{0}}}{\binom {-y_{0}}{x_{0}}}.}
3345:
4931:
5964:
7923:
5912:
5863:
682:
7330:
2732:
7779:
contacts the transformed third circle. Re-inversion produces the corresponding solutions to the original problem.
7731:
showed in the early 19th century that these six points lie on four lines, each line having three collinear points.
7534:
Further, the notion of bitangent lines can be extended to circles with negative radius (the same locus of points,
7217:
191:
No tangent line can be drawn through a point within a circle, since any such line must be a secant line. However,
8168:
121:
584:
3680:
355:
8074:
7857:
7537:
6474:
3667:
of any number of sides; in addition, for every number of polygon sides there are an infinite number of non-
2036:
7745:
3268:
2271:
212:
133:
109:
2660:
1667:, respectively. But only a tangent line is perpendicular to the radial line. Hence, the two lines from
7868:
7856:, lines are viewed as circles through a point "at infinity" and for any line and any circle, there is a
7755:
7474:
3668:
1924:
349:
8012:
7228:
6620:{\displaystyle {\begin{pmatrix}R&\mp {\sqrt {1-R^{2}}}\\\pm {\sqrt {1-R^{2}}}&R\end{pmatrix}}}
5692:{\displaystyle {\begin{aligned}ax_{1}+by_{1}+c&=r_{1}\\ax_{2}+by_{2}+c&=r_{2}\\\end{aligned}}}
4255:
2207:
8061:
7415:
2099:
5520:
8165:
7861:
7834:
7830:
7281:
6128:{\displaystyle X={\frac {\Delta x}{d}},\quad Y={\frac {\Delta y}{d}},\quad R={\frac {\Delta r}{d}}}
3597:
2151:
300:
153:
145:
7160:
3464:
1984:
8252:
7853:
7820:
7808:
7775:
7771:
3648:
137:
125:
59:
31:
7849:
7646:
5473:
4303:
3507:
1547:
3558:
8292:
8273:
8210:
7724:
4921:
4292:
4280:
4260:
4240:
578:
1886:
50:, never entering the circle's interior. Tangent lines to circles form the subject of several
8319:
8244:
7272:
7200:
is perpendicular to the radii, and that the tangent points lie on their respective circles.
5004:
depending upon the need to find distances from the centers of the nearer or farther circle,
4288:
3695:
3457:
307:
216:
164:
71:
47:
8295:
17:
7213:
4284:
3664:
254:
have a conjugate relationship to one another, which has been generalized into the idea of
234:
141:
39:
1645:; since both are inscribed in a semicircle, they are perpendicular to the line segments
8276:
8151:
7872:
7826:
7816:
3550:
255:
5053:
Inner tangent. The external tangent lines pass through the internal homothetic center.
8313:
8256:
7728:
4244:
3703:
1524:
90:
78:
3790:{\displaystyle {\overline {AB}}+{\overline {CD}}={\overline {BC}}+{\overline {DA}}.}
8229:
8215:
7791:
The concept of a tangent line and tangent point can be generalized to a pole point
7629:
7622:
6515:, equivalently the direction of rotation), and the above equations are rotation of
4270:
For two circles, there are generally four distinct lines that are tangent to both (
1705:
159:
8181:
1539:
Construction of a tangent to a given circle (black) from a given exterior point (
148:
of the tangent line and circle, even though the line and circle may be deformed.
6221:{\displaystyle {\begin{aligned}aX+bY&=R,\\a^{2}+b^{2}&=1;\end{aligned}}}
5169:
4894:
2196:
1535:
227:
to any two intersection points of the circle with a secant line passing through
117:
86:
8248:
7895:
5291:
7774:
two of the three given circles until they just touch, i.e., are tangent. An
3686:
is a closed figure of four straight sides that are tangent to a given circle
8300:
8281:
8237:
BSHM Bulletin: Journal of the
British Society for the History of Mathematics
7838:
7787:
5049:
4271:
4054:
But each side of the quadrilateral is composed of two such tangent segments
8146:
7875:. The parametric representation of the unit hyperbola via radius vector is
7212:
lines can also be generalized to circles with negative or zero radius. The
3799:
262:
outside the circle and the secant line joining its two points of tangency.
6135:
to simplify equation (1), resulting in the following system of equations:
7723:). Since each pair of circles has two homothetic centers, there are six
4920:
and the point where the two outer tangents of the two circles intersect (
3660:
3656:
129:
3706:, the sums of opposite sides of any such quadrilateral are equal, i.e.,
3652:
63:
51:
7867:
Radius and tangent line are perpendicular at a point of a circle, and
6746:
5133:
4902:
The distances between the centers of the nearer and farther circles,
242:
between a chord and a tangent is half the arc belonging to the chord.
82:
67:
43:
7157:
These equations express that the tangent line, which is parallel to
4243:, and that every rhombus has an inscribed circle, whereas a general
1611:
and the new circle are the tangent points for lines passing through
7841:
to this quartic curve. A generic quartic curve has 28 bitangents.
7786:
7744:
7627:
The internal and external tangent lines are useful in solving the
6745:
5290:
5168:
5048:
4893:
4888:
4302:
4254:
3798:
3655:
each of whose sides is tangent to a particular circle, called its
2195:
1534:
233:
158:
144:. In technical language, these transformations do not change the
5853:{\displaystyle \Delta x=x_{2}-x_{1},\quad \Delta y=y_{2}-y_{1}}
3265:
lies on the circle and there is just one tangent with equation
1566:
A circle is drawn centered on the midpoint of the line segment
85:
to that point, theorems involving tangent lines often involve
6663:
is the tangent line to the right of the circles looking from
6636:
is the tangent line to the right of the circles looking from
6464:
is the angle between the line of centers and a tangent line.
8204:
5256:
to a point. Two radial lines may be drawn from the center
4928:
respectively can be found out using similarity as follows:
4347:
is the outer tangent between the two circles. Given points
2092:
is a normal vector of the line. The tangent intersects the
773:
We begin by taking the implicit derivative with respect to
3445:{\displaystyle x_{1}x+y_{1}y=r^{2},\ x_{2}x+y_{2}y=r^{2}.}
2796:
and the unit base vectors by the orthogonal unit vectors
8169:"When A Quadrilateral Is Inscriptible?" at Cut-the-knot
6772:
for the four lines tangent to two circles with centers
5768:{\displaystyle a\Delta x+b\Delta y=\Delta r,\qquad (1)}
4274:) – if the two circles are outside each other – but in
3847:). The symmetric tangent segments about each point of
1712:
Draw any three different lines through the given point
568:{\displaystyle (x-x_{1})(x_{1}-a)+(y-y_{1})(y_{1}-b)=0}
258:. The same reciprocal relation exists between a point
8097:
8017:
7928:
7837:, and the external and internal tangent lines are the
6685:
The above assumes each circle has positive radius. If
6546:
6022:{\textstyle d={\sqrt {(\Delta x)^{2}+(\Delta y)^{2}}}}
5967:
2802:
2735:
207:
of the circle. Thus the lengths of the segments from
8077:
8015:
7926:
7540:
7477:
7418:
7333:
7284:
7231:
7163:
6816:
6540:
6477:
6241:
6141:
6053:
5915:
5866:
5781:
5721:
5572:
5523:
5476:
4934:
4664:
4431:
4062:
3857:
3714:
3600:
3561:
3510:
3467:
3348:
3271:
2992:
2663:
2486:
2334:
2274:
2210:
2154:
2102:
2039:
1987:
1927:
1889:
1512:, the center of the circle, through the radial point
1044:
785:
685:
587:
467:
358:
7992:
The radius and tangent are hyperbolic orthogonal at
7985:{\displaystyle {\tfrac {dp}{da}}=(\sinh a,\cosh a).}
5333:. Using the method above, two lines are drawn from
5211:. Using the method above, two lines are drawn from
195:
tangent lines can be drawn to a circle from a point
7770:special case). To accomplish this, it suffices to
4644:can be computed using basic trigonometry. You have
4307:
Finding outer tangent. Two circles' outer tangents.
3675:
Tangent quadrilateral theorem and inscribed circles
581:of the circle. Say that the circle has equation of
223:. This power equals the product of distances from
8147:"Finding tangents to a circle with a straightedge"
8123:
8040:
7984:
7591:
7519:
7460:
7401:
7316:
7263:
7192:
7147:
6619:
6503:
6408:
6220:
6127:
6021:
5950:
5901:
5852:
5767:
5691:
5558:
5509:
4974:
4879:
4640:notate the radii of the two circles and the angle
4622:
4223:
4046:
3789:
3632:
3586:
3538:
3496:
3444:
3319:
3216:
2969:
2788:
2708:
2647:
2472:
2320:
2251:
2186:
2140:
2084:
2021:
1969:
1913:
1473:
1012:
766:{\displaystyle (x_{1}-a)^{2}+(y_{1}-b)^{2}=r^{2}.}
765:
653:
567:
421:
8230:"The tangency problem of Apollonius: three looks"
7402:{\displaystyle |r_{1}-r_{2}|<d<r_{1}+r_{2}}
6808:are given by solving the simultaneous equations:
5369:. Two radial lines may be drawn from the center
3205:
3173:
3107:
3080:
3039:
2996:
2633:
2620:
2559:
2546:
2510:
2490:
54:, and play an important role in many geometrical
5715:by subtracting the first from the second yields
5470:respectively. Expressing a line by the equation
4419:can easily be calculated with help of the angle
2789:{\textstyle d_{0}={\sqrt {x_{0}^{2}+y_{0}^{2}}}}
661:and we are finding the slope of tangent line at
27:Line which touches a circle at exactly one point
8050:are reflections of each other in the asymptote
7641:Tangent lines to three circles: Monge's theorem
215:, the square of this tangent length equals the
187:, the length of the tangent line segment (red).
7471:If one circle is completely inside the other (
7412:If the circles touch internally at one point (
4291:, the external homothetic center lies at the
2958:
2926:
2872:
2845:
211:to the two tangent points are equal. By the
8:
8060:of the unit hyperbola. When interpreted as
7909:points in the direction of tangent line at
4295:corresponding to the slope of these lines.
348:Suppose that the equation of the circle in
4259:The external (above) and internal (below)
1855:This line meets the circle at two points,
654:{\displaystyle (x-a)^{2}+(y-b)^{2}=r^{2},}
8096:
8076:
8016:
8014:
7927:
7925:
7811:of each other with respect to the circle.
7580:
7558:
7545:
7539:
7512:
7506:
7493:
7484:
7476:
7453:
7447:
7434:
7425:
7417:
7393:
7380:
7362:
7356:
7343:
7334:
7332:
7308:
7295:
7283:
7255:
7242:
7230:
7181:
7168:
7162:
7135:
7130:
7110:
7097:
7078:
7065:
7048:
7043:
7023:
7010:
6991:
6978:
6948:
6935:
6916:
6903:
6873:
6860:
6841:
6828:
6817:
6815:
6596:
6584:
6569:
6557:
6541:
6539:
6493:
6481:
6476:
6393:
6377:
6358:
6332:
6320:
6283:
6271:
6242:
6240:
6195:
6182:
6142:
6140:
6110:
6085:
6060:
6052:
6011:
5989:
5974:
5966:
5942:
5929:
5914:
5893:
5880:
5865:
5844:
5831:
5808:
5795:
5780:
5720:
5679:
5656:
5640:
5623:
5600:
5584:
5573:
5571:
5541:
5528:
5522:
5475:
4975:{\displaystyle {\frac {dr}{r_{1}-r_{2}}}}
4963:
4950:
4935:
4933:
4860:
4850:
4837:
4821:
4811:
4798:
4777:
4738:
4725:
4720:
4710:
4697:
4692:
4682:
4665:
4663:
4595:
4578:
4549:
4532:
4503:
4486:
4457:
4440:
4432:
4430:
4166:
4148:
4144:
4086:
4068:
4063:
4061:
4016:
3998:
3971:
3953:
3925:
3907:
3880:
3862:
3858:
3856:
3769:
3751:
3733:
3715:
3713:
3621:
3608:
3599:
3569:
3560:
3518:
3509:
3488:
3475:
3466:
3433:
3417:
3401:
3385:
3369:
3353:
3347:
3308:
3292:
3276:
3270:
3204:
3197:
3186:
3172:
3170:
3162:
3149:
3144:
3138:
3130:
3125:
3116:
3106:
3099:
3089:
3079:
3077:
3069:
3064:
3054:
3048:
3038:
3027:
3023:
3009:
3005:
2995:
2993:
2991:
2957:
2950:
2939:
2925:
2923:
2915:
2906:
2897:
2886:
2885:
2871:
2864:
2854:
2844:
2842:
2834:
2825:
2816:
2805:
2804:
2801:
2778:
2773:
2760:
2755:
2749:
2740:
2734:
2697:
2684:
2668:
2662:
2632:
2619:
2617:
2609:
2596:
2591:
2585:
2577:
2568:
2558:
2545:
2543:
2535:
2525:
2519:
2509:
2502:
2489:
2487:
2485:
2459:
2446:
2441:
2435:
2427:
2418:
2404:
2391:
2385:
2373:
2357:
2347:
2341:
2333:
2309:
2283:
2279:
2273:
2231:
2215:
2209:
2175:
2162:
2153:
2123:
2107:
2101:
2076:
2041:
2040:
2038:
2010:
1986:
1958:
1945:
1932:
1926:
1888:
1442:
1426:
1395:
1379:
1347:
1331:
1293:
1277:
1245:
1227:
1220:
1205:
1185:
1156:
1138:
1131:
1125:
1112:
1087:
1069:
1062:
1045:
1043:
991:
973:
966:
936:
902:
853:
836:
811:
790:
786:
784:
754:
741:
725:
709:
693:
684:
642:
629:
604:
586:
544:
528:
497:
481:
466:
441:. Then the tangent line of the circle at
422:{\displaystyle (x-a)^{2}+(y-b)^{2}=r^{2}}
413:
400:
375:
357:
8124:{\displaystyle jp(a)={\tfrac {dp}{da}}.}
7327:If the circles intersect in two points (
3504:describes the circle inversion of point
8138:
7278:If they touch externally at one point (
7225:If the circles are outside each other (
6750:Finding outer tangent. Circle tangents.
2720:-axis: In the vector form one replaces
1921:be a point of the circle with equation
6712:will lie to the left of each line and
6235:) for the two external tangent lines:
1484:Compass and straightedge constructions
7592:{\displaystyle x^{2}+y^{2}=(-r)^{2},}
6504:{\displaystyle \pm {\sqrt {1-R^{2}}}}
5067:
4275:
3663:is a tangential polygon, as is every
2204:Conversely, if one starts with point
2085:{\displaystyle {\vec {OP}}=(a,b)^{T}}
7:
7741:Special cases of Apollonius' problem
7678:, there are three pairs of circles (
7608:the internal center is not defined.
5951:{\displaystyle \Delta r=r_{2}+r_{1}}
5902:{\displaystyle \Delta r=r_{2}-r_{1}}
5041:between the centers of two circles.
3342:there are 2 tangents with equations
3320:{\displaystyle x_{0}x+y_{0}y=r^{2}.}
2321:{\displaystyle P_{1/2}=(a,b_{\pm })}
1704:external to the circle using only a
1500:on the circumference of the circle:
1488:It is relatively straightforward to
269:is exterior to a circle with center
5096:be the centers of the two circles,
2709:{\displaystyle P_{0}=(x_{0},y_{0})}
7520:{\displaystyle d<|r_{1}-r_{2}|}
6754:In general the points of tangency
6228:solve these to get two solutions (
6113:
6088:
6063:
6001:
5979:
5916:
5867:
5818:
5782:
5746:
5737:
5725:
3177:
3084:
3000:
2930:
2849:
2624:
2550:
2494:
2268:meet the circle at the two points
1970:{\displaystyle x^{2}+y^{2}=r^{2}.}
1582:is again the center of the circle
25:
8041:{\displaystyle {\tfrac {dp}{da}}}
7795:and its corresponding polar line
6431:is the unit vector pointing from
5566:then a bitangent line satisfies:
5295:Construction of the inner tangent
5173:Construction of the outer tangent
577:This can be proved by taking the
313:is drawn from the tangency point
7264:{\displaystyle d>r_{1}+r_{2}}
4311:The red line joining the points
4279:lines intersect in the external
2986:touch the circle at the points
2977:Then the tangents through point
2252:{\displaystyle P_{0}=(x_{0},0),}
1716:that intersect the circle twice.
273:, and if the tangent lines from
7461:{\displaystyle d=|r_{1}-r_{2}|}
6103:
6078:
5817:
5755:
3997:
3906:
2368:
2141:{\displaystyle P_{0}=(x_{0},0)}
1496:tangent to a circle at a point
8296:"Tangent lines to two circles"
8090:
8084:
7976:
7952:
7577:
7567:
7513:
7485:
7454:
7426:
7363:
7335:
7116:
7090:
7084:
7058:
7029:
7003:
6997:
6971:
6954:
6928:
6922:
6896:
6879:
6853:
6847:
6821:
6399:
6367:
6008:
5998:
5986:
5976:
5762:
5756:
5559:{\displaystyle a^{2}+b^{2}=1,}
5378:through the tangent points on
5265:through the tangent points on
4857:
4830:
4818:
4791:
4214:
4202:
4196:
4184:
4134:
4122:
4116:
4104:
3581:
3562:
3530:
3511:
2891:
2810:
2703:
2677:
2315:
2296:
2259:than the two tangents through
2243:
2224:
2135:
2116:
2073:
2060:
2051:
1908:
1896:
1454:
1435:
1432:
1413:
1407:
1388:
1385:
1366:
1359:
1340:
1337:
1318:
1305:
1286:
1283:
1264:
1217:
1198:
899:
887:
878:
866:
833:
820:
808:
792:
738:
718:
706:
686:
626:
613:
601:
588:
556:
537:
534:
515:
509:
490:
487:
468:
397:
384:
372:
359:
156:about the axis of the radius.
1:
8277:"Tangent lines to one circle"
7651:For three circles denoted by
7317:{\displaystyle d=r_{1}+r_{2}}
5404:Let the circles have centres
4898:Outer tangents to two circles
3633:{\displaystyle xx_{0}=r^{2}.}
2187:{\displaystyle ax_{0}=r^{2}.}
1774:be the point where the lines
1700:the tangent lines to a point
1554:the tangent lines to a point
112:the circle at a single point
7275:, there are four bitangents.
7193:{\displaystyle t_{2}-t_{1},}
4251:Tangent lines to two circles
4176:
4158:
4096:
4078:
4026:
4008:
3981:
3963:
3935:
3917:
3890:
3872:
3779:
3761:
3743:
3725:
3690:. Equivalently, the circle
3497:{\displaystyle ax_{0}=r^{2}}
2033:lies on both the curves and
2022:{\displaystyle ax+by=r^{2},}
1615:, by the following argument.
8071:), the two numbers satisfy
7827:pole points and polar lines
6534:using the rotation matrix:
5351:to a point while expanding
4891:the 2-argument arctangent.
1866:The tangents are the lines
277:touch the circle at points
256:pole points and polar lines
97:Tangent lines to one circle
18:Tangent between two circles
8336:
8202:Libeskind, Shlomo (2007),
7814:
7738:
7644:
7620:
6511:(depending on the sign of
5909:for the outer tangent or
5510:{\displaystyle ax+by+c=0,}
3539:{\displaystyle (x_{0},0).}
1689:are tangent to the circle
8249:10.1080/17498430601148911
5152:; in other words, circle
3587:{\displaystyle (x_{0},0)}
167:, the product of lengths
3803:Tangential quadrilateral
3681:tangential quadrilateral
2480:Written in vector form:
2200:Tangents through a point
1641:are radii of the circle
1589:The intersection points
181:equals to the square of
165:power-of-a-point theorem
36:tangent line to a circle
32:Euclidean plane geometry
7640:
5958:for the inner tangent.
5517:with the normalization
1914:{\displaystyle P=(a,b)}
1558:external to the circle
459:has Cartesian equation
8125:
8042:
7986:
7812:
7754:Many special cases of
7751:
7593:
7521:
7462:
7403:
7318:
7265:
7194:
7149:
6751:
6621:
6505:
6410:
6222:
6129:
6023:
5952:
5903:
5854:
5769:
5693:
5560:
5511:
5360:by a constant amount,
5296:
5238:by a constant amount,
5174:
5054:
4976:
4899:
4881:
4624:
4308:
4267:
4225:
4048:
3804:
3791:
3634:
3588:
3540:
3498:
3446:
3321:
3218:
2971:
2790:
2710:
2649:
2474:
2322:
2253:
2201:
2188:
2142:
2086:
2023:
1971:
1915:
1879:With analytic geometry
1544:
1475:
1014:
767:
655:
569:
423:
250:and the tangent point
243:
213:secant-tangent theorem
188:
8228:Kunkel, Paul (2007),
8126:
8062:split-complex numbers
8043:
7987:
7869:hyperbolic-orthogonal
7858:Möbius transformation
7790:
7748:
7735:Problem of Apollonius
7594:
7522:
7463:
7404:
7319:
7266:
7195:
7150:
6749:
6622:
6506:
6411:
6223:
6130:
6047:we can normalize by
6029:is the distance from
6024:
5953:
5904:
5855:
5770:
5694:
5561:
5512:
5324:is drawn centered on
5294:
5202:is drawn centered on
5172:
5052:
4977:
4897:
4882:
4625:
4306:
4258:
4233:proving the theorem.
4226:
4049:
3802:
3792:
3698:in the quadrilateral
3671:tangential polygons.
3635:
3589:
3541:
3499:
3447:
3322:
3219:
2972:
2791:
2711:
2650:
2475:
2323:
2254:
2199:
2189:
2143:
2087:
2024:
1972:
1916:
1538:
1476:
1015:
768:
656:
570:
424:
350:Cartesian coordinates
237:
162:
8075:
8013:
7924:
7756:Apollonius's problem
7538:
7475:
7416:
7331:
7282:
7229:
7161:
6814:
6538:
6475:
6239:
6139:
6051:
5965:
5913:
5864:
5779:
5719:
5570:
5521:
5474:
4932:
4662:
4429:
4060:
3855:
3712:
3598:
3559:
3508:
3465:
3346:
3269:
2990:
2800:
2733:
2661:
2484:
2332:
2272:
2208:
2152:
2100:
2037:
1985:
1925:
1887:
1844:Draw a line through
1671:and passing through
1042:
783:
683:
585:
465:
356:
217:power of the point P
203:to the center point
8184:. Whistleralley.com
8166:Alexander Bogomolny
7862:Lie sphere geometry
7835:quartic plane curve
7140:
7053:
4266:of the two circles.
3643:Tangential polygons
3555:The polar of point
3154:
3135:
3074:
2783:
2765:
2601:
2451:
579:implicit derivative
154:reflection symmetry
146:incidence structure
116:. For comparison,
8293:Weisstein, Eric W.
8274:Weisstein, Eric W.
8206:, pp. 110–112
8155:. August 15, 2015.
8121:
8116:
8038:
8036:
7982:
7947:
7871:at a point of the
7854:inversive geometry
7821:Inversive geometry
7813:
7752:
7750:within or without.
7725:homothetic centers
7589:
7517:
7458:
7399:
7314:
7261:
7190:
7145:
7143:
7126:
7039:
6752:
6617:
6611:
6501:
6406:
6404:
6218:
6216:
6125:
6019:
5948:
5899:
5850:
5765:
5689:
5687:
5556:
5507:
5387:; these intersect
5297:
5274:; these intersect
5175:
5074:Synthetic geometry
5055:
4972:
4900:
4877:
4875:
4620:
4618:
4309:
4268:
4221:
4219:
4044:
4042:
3805:
3787:
3649:tangential polygon
3630:
3584:
3536:
3494:
3442:
3317:
3240:no tangents exist.
3214:
3140:
3121:
3060:
2967:
2786:
2769:
2751:
2706:
2645:
2587:
2470:
2437:
2318:
2249:
2202:
2184:
2138:
2082:
2019:
1967:
1911:
1696:Another method to
1619:The line segments
1572:, having diameter
1545:
1471:
1469:
1010:
1008:
763:
651:
565:
419:
344:Cartesian equation
317:of exterior point
244:
189:
8182:"Tangent circles"
8115:
8035:
7946:
6602:
6575:
6499:
6338:
6289:
6123:
6098:
6073:
6017:
5400:Analytic geometry
5287:Internal tangents
5165:External tangents
4970:
4922:homothetic center
4867:
4866:
4685:
4293:point at infinity
4281:homothetic center
4261:homothetic center
4179:
4161:
4099:
4081:
4029:
4011:
3984:
3966:
3938:
3920:
3893:
3875:
3782:
3764:
3746:
3728:
3396:
3203:
3168:
3136:
3105:
3075:
3037:
2956:
2921:
2894:
2883:
2870:
2840:
2813:
2784:
2641:
2631:
2615:
2583:
2557:
2541:
2508:
2465:
2433:
2410:
2363:
2054:
1258:
1169:
1100:
1004:
954:
920:
796:
795:
246:The tangent line
42:that touches the
16:(Redirected from
8327:
8306:
8305:
8287:
8286:
8260:
8259:
8234:
8225:
8219:
8207:
8199:
8193:
8192:
8190:
8189:
8177:
8171:
8163:
8157:
8156:
8143:
8130:
8128:
8127:
8122:
8117:
8114:
8106:
8098:
8070:
8059:
8049:
8047:
8045:
8044:
8039:
8037:
8034:
8026:
8018:
8006:
7995:
7991:
7989:
7988:
7983:
7948:
7945:
7937:
7929:
7919:
7908:
7893:
7806:
7802:
7798:
7794:
7722:
7707:
7692:
7677:
7668:
7659:
7598:
7596:
7595:
7590:
7585:
7584:
7563:
7562:
7550:
7549:
7526:
7524:
7523:
7518:
7516:
7511:
7510:
7498:
7497:
7488:
7467:
7465:
7464:
7459:
7457:
7452:
7451:
7439:
7438:
7429:
7408:
7406:
7405:
7400:
7398:
7397:
7385:
7384:
7366:
7361:
7360:
7348:
7347:
7338:
7323:
7321:
7320:
7315:
7313:
7312:
7300:
7299:
7273:general position
7270:
7268:
7267:
7262:
7260:
7259:
7247:
7246:
7214:degenerate cases
7207:Degenerate cases
7199:
7197:
7196:
7191:
7186:
7185:
7173:
7172:
7154:
7152:
7151:
7146:
7144:
7139:
7134:
7115:
7114:
7102:
7101:
7083:
7082:
7070:
7069:
7052:
7047:
7028:
7027:
7015:
7014:
6996:
6995:
6983:
6982:
6953:
6952:
6940:
6939:
6921:
6920:
6908:
6907:
6878:
6877:
6865:
6864:
6846:
6845:
6833:
6832:
6807:
6798:
6789:
6780:
6771:
6762:
6734:
6727:
6720:
6711:
6702:
6694:is positive and
6693:
6680:
6671:
6662:
6653:
6644:
6635:
6626:
6624:
6623:
6618:
6616:
6615:
6603:
6601:
6600:
6585:
6576:
6574:
6573:
6558:
6533:
6526:
6514:
6510:
6508:
6507:
6502:
6500:
6498:
6497:
6482:
6470:
6463:
6459:
6452:
6448:
6439:
6430:
6415:
6413:
6412:
6407:
6405:
6398:
6397:
6382:
6381:
6363:
6362:
6339:
6337:
6336:
6321:
6290:
6288:
6287:
6272:
6234:
6227:
6225:
6224:
6219:
6217:
6200:
6199:
6187:
6186:
6134:
6132:
6131:
6126:
6124:
6119:
6111:
6099:
6094:
6086:
6074:
6069:
6061:
6046:
6037:
6028:
6026:
6025:
6020:
6018:
6016:
6015:
5994:
5993:
5975:
5957:
5955:
5954:
5949:
5947:
5946:
5934:
5933:
5908:
5906:
5905:
5900:
5898:
5897:
5885:
5884:
5859:
5857:
5856:
5851:
5849:
5848:
5836:
5835:
5813:
5812:
5800:
5799:
5774:
5772:
5771:
5766:
5714:
5698:
5696:
5695:
5690:
5688:
5684:
5683:
5661:
5660:
5645:
5644:
5628:
5627:
5605:
5604:
5589:
5588:
5565:
5563:
5562:
5557:
5546:
5545:
5533:
5532:
5516:
5514:
5513:
5508:
5469:
5460:
5451:
5427:
5395:
5386:
5377:
5368:
5359:
5350:
5341:
5332:
5323:
5307:
5282:
5273:
5264:
5255:
5247:, which shrinks
5246:
5237:
5228:
5219:
5210:
5201:
5185:
5160:
5151:
5131:
5122:
5113:
5104:
5095:
5086:
5068:degenerate cases
5040:
5026:is the distance
5025:
5021:
5012:
5003:
4994:
4985:
4981:
4979:
4978:
4973:
4971:
4969:
4968:
4967:
4955:
4954:
4944:
4936:
4927:
4919:
4910:
4886:
4884:
4883:
4878:
4876:
4872:
4868:
4865:
4864:
4855:
4854:
4842:
4841:
4826:
4825:
4816:
4815:
4803:
4802:
4790:
4789:
4778:
4749:
4745:
4744:
4743:
4742:
4730:
4729:
4716:
4715:
4714:
4702:
4701:
4686:
4683:
4657:
4643:
4639:
4635:
4629:
4627:
4626:
4621:
4619:
4600:
4599:
4583:
4582:
4554:
4553:
4537:
4536:
4508:
4507:
4491:
4490:
4462:
4461:
4445:
4444:
4422:
4418:
4400:
4382:
4364:
4346:
4328:
4289:projective plane
4276:degenerate cases
4265:
4230:
4228:
4227:
4222:
4220:
4180:
4175:
4167:
4162:
4157:
4149:
4145:
4100:
4095:
4087:
4082:
4077:
4069:
4066:
4053:
4051:
4050:
4045:
4043:
4030:
4025:
4017:
4012:
4007:
3999:
3985:
3980:
3972:
3967:
3962:
3954:
3939:
3934:
3926:
3921:
3916:
3908:
3894:
3889:
3881:
3876:
3871:
3863:
3850:
3846:
3845:
3840:
3836:
3835:
3830:
3826:
3825:
3820:
3816:
3815:
3810:
3796:
3794:
3793:
3788:
3783:
3778:
3770:
3765:
3760:
3752:
3747:
3742:
3734:
3729:
3724:
3716:
3701:
3693:
3689:
3685:
3639:
3637:
3636:
3631:
3626:
3625:
3613:
3612:
3593:
3591:
3590:
3585:
3574:
3573:
3545:
3543:
3542:
3537:
3523:
3522:
3503:
3501:
3500:
3495:
3493:
3492:
3480:
3479:
3458:circle inversion
3451:
3449:
3448:
3443:
3438:
3437:
3422:
3421:
3406:
3405:
3394:
3390:
3389:
3374:
3373:
3358:
3357:
3341:
3326:
3324:
3323:
3318:
3313:
3312:
3297:
3296:
3281:
3280:
3264:
3255:
3239:
3223:
3221:
3220:
3215:
3210:
3209:
3208:
3202:
3201:
3192:
3191:
3190:
3176:
3169:
3167:
3166:
3153:
3148:
3139:
3137:
3134:
3129:
3117:
3112:
3111:
3110:
3104:
3103:
3094:
3093:
3083:
3076:
3073:
3068:
3059:
3058:
3049:
3044:
3043:
3042:
3036:
3035:
3031:
3018:
3017:
3013:
2999:
2985:
2976:
2974:
2973:
2968:
2963:
2962:
2961:
2955:
2954:
2945:
2944:
2943:
2929:
2922:
2920:
2919:
2907:
2902:
2901:
2896:
2895:
2887:
2881:
2877:
2876:
2875:
2869:
2868:
2859:
2858:
2848:
2841:
2839:
2838:
2826:
2821:
2820:
2815:
2814:
2806:
2795:
2793:
2792:
2787:
2785:
2782:
2777:
2764:
2759:
2750:
2745:
2744:
2729:by the distance
2728:
2719:
2716:lies not on the
2715:
2713:
2712:
2707:
2702:
2701:
2689:
2688:
2673:
2672:
2654:
2652:
2651:
2646:
2639:
2638:
2637:
2636:
2623:
2616:
2614:
2613:
2600:
2595:
2586:
2584:
2582:
2581:
2569:
2564:
2563:
2562:
2549:
2542:
2540:
2539:
2530:
2529:
2520:
2515:
2514:
2513:
2507:
2506:
2493:
2479:
2477:
2476:
2471:
2466:
2464:
2463:
2450:
2445:
2436:
2434:
2432:
2431:
2419:
2411:
2409:
2408:
2396:
2395:
2386:
2378:
2377:
2364:
2362:
2361:
2352:
2351:
2342:
2327:
2325:
2324:
2319:
2314:
2313:
2292:
2291:
2287:
2267:
2258:
2256:
2255:
2250:
2236:
2235:
2220:
2219:
2193:
2191:
2190:
2185:
2180:
2179:
2167:
2166:
2147:
2145:
2144:
2139:
2128:
2127:
2112:
2111:
2095:
2091:
2089:
2088:
2083:
2081:
2080:
2056:
2055:
2050:
2042:
2032:
2028:
2026:
2025:
2020:
2015:
2014:
1980:
1976:
1974:
1973:
1968:
1963:
1962:
1950:
1949:
1937:
1936:
1920:
1918:
1917:
1912:
1873:
1869:
1862:
1858:
1851:
1847:
1840:
1825:
1810:
1803:
1788:
1773:
1766:
1762:
1715:
1703:
1692:
1688:
1679:
1670:
1666:
1662:
1655:
1651:
1644:
1640:
1636:
1629:
1625:
1614:
1610:
1606:
1597:
1585:
1581:
1577:
1576:
1571:
1570:
1561:
1557:
1542:
1530:
1522:
1515:
1511:
1507:
1499:
1495:
1480:
1478:
1477:
1472:
1470:
1447:
1446:
1431:
1430:
1400:
1399:
1384:
1383:
1352:
1351:
1336:
1335:
1298:
1297:
1282:
1281:
1259:
1257:
1250:
1249:
1239:
1232:
1231:
1221:
1210:
1209:
1190:
1189:
1170:
1168:
1161:
1160:
1150:
1143:
1142:
1132:
1130:
1129:
1117:
1116:
1101:
1099:
1092:
1091:
1081:
1074:
1073:
1063:
1035:
1019:
1017:
1016:
1011:
1009:
1005:
1003:
996:
995:
985:
978:
977:
967:
955:
953:
945:
937:
921:
919:
911:
903:
858:
857:
841:
840:
816:
815:
797:
791:
776:
772:
770:
769:
764:
759:
758:
746:
745:
730:
729:
714:
713:
698:
697:
678:
660:
658:
657:
652:
647:
646:
634:
633:
609:
608:
574:
572:
571:
566:
549:
548:
533:
532:
502:
501:
486:
485:
458:
440:
428:
426:
425:
420:
418:
417:
405:
404:
380:
379:
339:
328:
320:
316:
312:
298:
291:
284:
280:
276:
272:
268:
261:
253:
249:
241:
230:
226:
222:
210:
206:
202:
198:
186:
185:
180:
176:
175:
171:
115:
108:
104:
76:
21:
8335:
8334:
8330:
8329:
8328:
8326:
8325:
8324:
8310:
8309:
8291:
8290:
8272:
8271:
8268:
8263:
8232:
8227:
8226:
8222:
8201:
8200:
8196:
8187:
8185:
8179:
8178:
8174:
8164:
8160:
8145:
8144:
8140:
8136:
8107:
8099:
8073:
8072:
8065:
8051:
8027:
8019:
8011:
8010:
8008:
7997:
7993:
7938:
7930:
7922:
7921:
7910:
7899:
7876:
7823:
7815:Main articles:
7804:
7800:
7796:
7792:
7785:
7783:Generalizations
7743:
7737:
7721:
7715:
7709:
7706:
7700:
7694:
7691:
7685:
7679:
7676:
7670:
7667:
7661:
7658:
7652:
7649:
7647:Monge's theorem
7643:
7625:
7619:
7614:
7576:
7554:
7541:
7536:
7535:
7502:
7489:
7473:
7472:
7443:
7430:
7414:
7413:
7389:
7376:
7352:
7339:
7329:
7328:
7304:
7291:
7280:
7279:
7251:
7238:
7227:
7226:
7209:
7177:
7164:
7159:
7158:
7142:
7141:
7119:
7106:
7093:
7074:
7061:
7055:
7054:
7032:
7019:
7006:
6987:
6974:
6968:
6967:
6957:
6944:
6931:
6912:
6899:
6893:
6892:
6882:
6869:
6856:
6837:
6824:
6812:
6811:
6806:
6800:
6797:
6791:
6788:
6782:
6779:
6773:
6770:
6764:
6761:
6755:
6744:
6738:
6729:
6722:
6719:
6713:
6710:
6704:
6701:
6695:
6692:
6686:
6679:
6673:
6670:
6664:
6657:
6652:
6646:
6643:
6637:
6630:
6610:
6609:
6604:
6592:
6578:
6577:
6565:
6552:
6542:
6536:
6535:
6528:
6516:
6512:
6489:
6473:
6472:
6465:
6461:
6454:
6450:
6447:
6441:
6438:
6432:
6420:
6403:
6402:
6389:
6373:
6354:
6347:
6341:
6340:
6328:
6298:
6292:
6291:
6279:
6249:
6237:
6236:
6229:
6215:
6214:
6201:
6191:
6178:
6175:
6174:
6161:
6137:
6136:
6112:
6087:
6062:
6049:
6048:
6045:
6039:
6036:
6030:
6007:
5985:
5963:
5962:
5938:
5925:
5911:
5910:
5889:
5876:
5862:
5861:
5840:
5827:
5804:
5791:
5777:
5776:
5717:
5716:
5700:
5686:
5685:
5675:
5668:
5652:
5636:
5630:
5629:
5619:
5612:
5596:
5580:
5568:
5567:
5537:
5524:
5519:
5518:
5472:
5471:
5468:
5462:
5459:
5453:
5449:
5442:
5435:
5429:
5425:
5418:
5411:
5405:
5402:
5394:
5388:
5385:
5379:
5376:
5370:
5367:
5361:
5358:
5352:
5349:
5343:
5340:
5334:
5331:
5325:
5322:
5315:
5309:
5306:
5300:
5281:
5275:
5272:
5266:
5263:
5257:
5254:
5248:
5245:
5239:
5236:
5230:
5227:
5221:
5218:
5212:
5209:
5203:
5200:
5193:
5187:
5184:
5178:
5159:
5153:
5150:
5143:
5137:
5130:
5124:
5121:
5115:
5112:
5106:
5103:
5097:
5094:
5088:
5085:
5079:
5076:
5063:
5047:
5039:
5033:
5027:
5023:
5020:
5014:
5011:
5005:
5002:
4996:
4993:
4987:
4983:
4959:
4946:
4945:
4937:
4930:
4929:
4925:
4918:
4912:
4909:
4903:
4874:
4873:
4856:
4846:
4833:
4817:
4807:
4794:
4779:
4773:
4757:
4751:
4750:
4734:
4721:
4706:
4693:
4691:
4687:
4672:
4660:
4659:
4645:
4641:
4637:
4633:
4617:
4616:
4591:
4584:
4574:
4571:
4570:
4545:
4538:
4528:
4525:
4524:
4499:
4492:
4482:
4479:
4478:
4453:
4446:
4436:
4427:
4426:
4420:
4416:
4409:
4402:
4398:
4391:
4384:
4380:
4373:
4366:
4362:
4355:
4348:
4344:
4337:
4330:
4326:
4319:
4312:
4301:
4263:
4253:
4218:
4217:
4168:
4150:
4146:
4138:
4137:
4088:
4070:
4058:
4057:
4041:
4040:
4018:
4000:
3995:
3973:
3955:
3950:
3949:
3927:
3909:
3904:
3882:
3864:
3853:
3852:
3848:
3843:
3842:
3838:
3833:
3832:
3828:
3823:
3822:
3818:
3813:
3812:
3808:
3771:
3753:
3735:
3717:
3710:
3709:
3699:
3691:
3687:
3683:
3677:
3665:regular polygon
3645:
3617:
3604:
3596:
3595:
3565:
3557:
3556:
3514:
3506:
3505:
3484:
3471:
3463:
3462:
3429:
3413:
3397:
3381:
3365:
3349:
3344:
3343:
3336:
3330:
3304:
3288:
3272:
3267:
3266:
3263:
3257:
3250:
3244:
3234:
3228:
3193:
3182:
3178:
3171:
3158:
3095:
3085:
3078:
3050:
3019:
3001:
2994:
2988:
2987:
2984:
2978:
2946:
2935:
2931:
2924:
2911:
2884:
2860:
2850:
2843:
2830:
2803:
2798:
2797:
2736:
2731:
2730:
2727:
2721:
2717:
2693:
2680:
2664:
2659:
2658:
2618:
2605:
2573:
2544:
2531:
2521:
2498:
2488:
2482:
2481:
2455:
2423:
2400:
2387:
2369:
2353:
2343:
2330:
2329:
2305:
2275:
2270:
2269:
2266:
2260:
2227:
2211:
2206:
2205:
2171:
2158:
2150:
2149:
2119:
2103:
2098:
2097:
2096:-axis at point
2093:
2072:
2043:
2035:
2034:
2030:
2006:
1983:
1982:
1978:
1977:The tangent at
1954:
1941:
1928:
1923:
1922:
1885:
1884:
1881:
1871:
1867:
1860:
1856:
1849:
1845:
1839:
1833:
1827:
1824:
1818:
1812:
1808:
1802:
1796:
1790:
1787:
1781:
1775:
1771:
1764:
1761:
1754:
1747:
1740:
1733:
1726:
1720:
1713:
1701:
1690:
1687:
1681:
1678:
1672:
1668:
1665:
1658:
1657:
1654:
1647:
1646:
1642:
1639:
1632:
1631:
1628:
1621:
1620:
1612:
1608:
1605:
1599:
1596:
1590:
1583:
1579:
1574:
1573:
1568:
1567:
1559:
1555:
1550:may be used to
1548:Thales' theorem
1540:
1528:
1520:
1513:
1509:
1505:
1497:
1493:
1486:
1468:
1467:
1457:
1438:
1422:
1391:
1375:
1363:
1362:
1343:
1327:
1308:
1289:
1273:
1261:
1260:
1241:
1240:
1223:
1222:
1201:
1191:
1181:
1172:
1171:
1152:
1151:
1134:
1133:
1121:
1108:
1083:
1082:
1065:
1064:
1052:
1040:
1039:
1023:
1007:
1006:
987:
986:
969:
968:
956:
946:
938:
933:
932:
922:
912:
904:
860:
859:
849:
842:
832:
807:
781:
780:
774:
750:
737:
721:
705:
689:
681:
680:
676:
669:
662:
638:
625:
600:
583:
582:
540:
524:
493:
477:
463:
462:
456:
449:
442:
430:
429:with center at
409:
396:
371:
354:
353:
346:
330:
322:
318:
314:
310:
303:(sum to 180°).
293:
286:
282:
278:
274:
270:
266:
259:
251:
247:
239:
228:
224:
220:
208:
204:
200:
196:
183:
182:
178:
173:
169:
168:
142:map projections
122:transformations
113:
106:
102:
101:A tangent line
99:
74:
46:at exactly one
28:
23:
22:
15:
12:
11:
5:
8333:
8331:
8323:
8322:
8312:
8311:
8308:
8307:
8288:
8267:
8266:External links
8264:
8262:
8261:
8220:
8194:
8172:
8158:
8152:Stack Exchange
8137:
8135:
8132:
8120:
8113:
8110:
8105:
8102:
8095:
8092:
8089:
8086:
8083:
8080:
8033:
8030:
8025:
8022:
7981:
7978:
7975:
7972:
7969:
7966:
7963:
7960:
7957:
7954:
7951:
7944:
7941:
7936:
7933:
7873:unit hyperbola
7817:Pole and polar
7799:. The points
7784:
7781:
7739:Main article:
7736:
7733:
7719:
7713:
7704:
7698:
7689:
7683:
7674:
7665:
7656:
7645:Main article:
7642:
7639:
7635:pulley problem
7621:Main article:
7618:
7615:
7613:
7610:
7588:
7583:
7579:
7575:
7572:
7569:
7566:
7561:
7557:
7553:
7548:
7544:
7529:
7528:
7515:
7509:
7505:
7501:
7496:
7492:
7487:
7483:
7480:
7469:
7456:
7450:
7446:
7442:
7437:
7433:
7428:
7424:
7421:
7410:
7396:
7392:
7388:
7383:
7379:
7375:
7372:
7369:
7365:
7359:
7355:
7351:
7346:
7342:
7337:
7325:
7311:
7307:
7303:
7298:
7294:
7290:
7287:
7276:
7258:
7254:
7250:
7245:
7241:
7237:
7234:
7218:multiplicities
7208:
7205:
7189:
7184:
7180:
7176:
7171:
7167:
7138:
7133:
7129:
7125:
7122:
7120:
7118:
7113:
7109:
7105:
7100:
7096:
7092:
7089:
7086:
7081:
7077:
7073:
7068:
7064:
7060:
7057:
7056:
7051:
7046:
7042:
7038:
7035:
7033:
7031:
7026:
7022:
7018:
7013:
7009:
7005:
7002:
6999:
6994:
6990:
6986:
6981:
6977:
6973:
6970:
6969:
6966:
6963:
6960:
6958:
6956:
6951:
6947:
6943:
6938:
6934:
6930:
6927:
6924:
6919:
6915:
6911:
6906:
6902:
6898:
6895:
6894:
6891:
6888:
6885:
6883:
6881:
6876:
6872:
6868:
6863:
6859:
6855:
6852:
6849:
6844:
6840:
6836:
6831:
6827:
6823:
6820:
6819:
6804:
6795:
6786:
6777:
6768:
6759:
6743:
6740:
6717:
6708:
6703:negative then
6699:
6690:
6683:
6682:
6677:
6668:
6655:
6650:
6641:
6614:
6608:
6605:
6599:
6595:
6591:
6588:
6583:
6580:
6579:
6572:
6568:
6564:
6561:
6556:
6553:
6551:
6548:
6547:
6545:
6496:
6492:
6488:
6485:
6480:
6445:
6436:
6401:
6396:
6392:
6388:
6385:
6380:
6376:
6372:
6369:
6366:
6361:
6357:
6353:
6350:
6348:
6346:
6343:
6342:
6335:
6331:
6327:
6324:
6319:
6316:
6313:
6310:
6307:
6304:
6301:
6299:
6297:
6294:
6293:
6286:
6282:
6278:
6275:
6270:
6267:
6264:
6261:
6258:
6255:
6252:
6250:
6248:
6245:
6244:
6213:
6210:
6207:
6204:
6202:
6198:
6194:
6190:
6185:
6181:
6177:
6176:
6173:
6170:
6167:
6164:
6162:
6160:
6157:
6154:
6151:
6148:
6145:
6144:
6122:
6118:
6115:
6109:
6106:
6102:
6097:
6093:
6090:
6084:
6081:
6077:
6072:
6068:
6065:
6059:
6056:
6043:
6034:
6014:
6010:
6006:
6003:
6000:
5997:
5992:
5988:
5984:
5981:
5978:
5973:
5970:
5945:
5941:
5937:
5932:
5928:
5924:
5921:
5918:
5896:
5892:
5888:
5883:
5879:
5875:
5872:
5869:
5847:
5843:
5839:
5834:
5830:
5826:
5823:
5820:
5816:
5811:
5807:
5803:
5798:
5794:
5790:
5787:
5784:
5764:
5761:
5758:
5754:
5751:
5748:
5745:
5742:
5739:
5736:
5733:
5730:
5727:
5724:
5682:
5678:
5674:
5671:
5669:
5667:
5664:
5659:
5655:
5651:
5648:
5643:
5639:
5635:
5632:
5631:
5626:
5622:
5618:
5615:
5613:
5611:
5608:
5603:
5599:
5595:
5592:
5587:
5583:
5579:
5576:
5575:
5555:
5552:
5549:
5544:
5540:
5536:
5531:
5527:
5506:
5503:
5500:
5497:
5494:
5491:
5488:
5485:
5482:
5479:
5466:
5457:
5447:
5440:
5433:
5423:
5416:
5409:
5401:
5398:
5392:
5383:
5374:
5365:
5356:
5347:
5338:
5329:
5320:
5313:
5304:
5289:
5288:
5279:
5270:
5261:
5252:
5243:
5234:
5225:
5216:
5207:
5198:
5191:
5182:
5167:
5166:
5157:
5148:
5141:
5128:
5119:
5110:
5101:
5092:
5083:
5075:
5072:
5062:
5059:
5046:
5043:
5037:
5031:
5018:
5009:
5000:
4991:
4966:
4962:
4958:
4953:
4949:
4943:
4940:
4916:
4907:
4871:
4863:
4859:
4853:
4849:
4845:
4840:
4836:
4832:
4829:
4824:
4820:
4814:
4810:
4806:
4801:
4797:
4793:
4788:
4785:
4782:
4776:
4772:
4769:
4766:
4763:
4760:
4758:
4756:
4753:
4752:
4748:
4741:
4737:
4733:
4728:
4724:
4719:
4713:
4709:
4705:
4700:
4696:
4690:
4681:
4678:
4675:
4673:
4671:
4668:
4667:
4615:
4612:
4609:
4606:
4603:
4598:
4594:
4590:
4587:
4585:
4581:
4577:
4573:
4572:
4569:
4566:
4563:
4560:
4557:
4552:
4548:
4544:
4541:
4539:
4535:
4531:
4527:
4526:
4523:
4520:
4517:
4514:
4511:
4506:
4502:
4498:
4495:
4493:
4489:
4485:
4481:
4480:
4477:
4474:
4471:
4468:
4465:
4460:
4456:
4452:
4449:
4447:
4443:
4439:
4435:
4434:
4414:
4407:
4396:
4389:
4378:
4371:
4360:
4353:
4342:
4335:
4324:
4317:
4300:
4297:
4252:
4249:
4216:
4213:
4210:
4207:
4204:
4201:
4198:
4195:
4192:
4189:
4186:
4183:
4178:
4174:
4171:
4165:
4160:
4156:
4153:
4147:
4143:
4140:
4139:
4136:
4133:
4130:
4127:
4124:
4121:
4118:
4115:
4112:
4109:
4106:
4103:
4098:
4094:
4091:
4085:
4080:
4076:
4073:
4067:
4065:
4039:
4036:
4033:
4028:
4024:
4021:
4015:
4010:
4006:
4003:
3996:
3994:
3991:
3988:
3983:
3979:
3976:
3970:
3965:
3961:
3958:
3952:
3951:
3948:
3945:
3942:
3937:
3933:
3930:
3924:
3919:
3915:
3912:
3905:
3903:
3900:
3897:
3892:
3888:
3885:
3879:
3874:
3870:
3867:
3861:
3860:
3786:
3781:
3777:
3774:
3768:
3763:
3759:
3756:
3750:
3745:
3741:
3738:
3732:
3727:
3723:
3720:
3676:
3673:
3644:
3641:
3629:
3624:
3620:
3616:
3611:
3607:
3603:
3583:
3580:
3577:
3572:
3568:
3564:
3551:pole and polar
3535:
3532:
3529:
3526:
3521:
3517:
3513:
3491:
3487:
3483:
3478:
3474:
3470:
3453:
3452:
3441:
3436:
3432:
3428:
3425:
3420:
3416:
3412:
3409:
3404:
3400:
3393:
3388:
3384:
3380:
3377:
3372:
3368:
3364:
3361:
3356:
3352:
3334:
3327:
3316:
3311:
3307:
3303:
3300:
3295:
3291:
3287:
3284:
3279:
3275:
3261:
3248:
3241:
3232:
3213:
3207:
3200:
3196:
3189:
3185:
3181:
3175:
3165:
3161:
3157:
3152:
3147:
3143:
3133:
3128:
3124:
3120:
3115:
3109:
3102:
3098:
3092:
3088:
3082:
3072:
3067:
3063:
3057:
3053:
3047:
3041:
3034:
3030:
3026:
3022:
3016:
3012:
3008:
3004:
2998:
2982:
2966:
2960:
2953:
2949:
2942:
2938:
2934:
2928:
2918:
2914:
2910:
2905:
2900:
2893:
2890:
2880:
2874:
2867:
2863:
2857:
2853:
2847:
2837:
2833:
2829:
2824:
2819:
2812:
2809:
2781:
2776:
2772:
2768:
2763:
2758:
2754:
2748:
2743:
2739:
2725:
2705:
2700:
2696:
2692:
2687:
2683:
2679:
2676:
2671:
2667:
2644:
2635:
2630:
2627:
2622:
2612:
2608:
2604:
2599:
2594:
2590:
2580:
2576:
2572:
2567:
2561:
2556:
2553:
2548:
2538:
2534:
2528:
2524:
2518:
2512:
2505:
2501:
2497:
2492:
2469:
2462:
2458:
2454:
2449:
2444:
2440:
2430:
2426:
2422:
2417:
2414:
2407:
2403:
2399:
2394:
2390:
2384:
2381:
2376:
2372:
2367:
2360:
2356:
2350:
2346:
2340:
2337:
2317:
2312:
2308:
2304:
2301:
2298:
2295:
2290:
2286:
2282:
2278:
2264:
2248:
2245:
2242:
2239:
2234:
2230:
2226:
2223:
2218:
2214:
2183:
2178:
2174:
2170:
2165:
2161:
2157:
2137:
2134:
2131:
2126:
2122:
2118:
2115:
2110:
2106:
2079:
2075:
2071:
2068:
2065:
2062:
2059:
2053:
2049:
2046:
2018:
2013:
2009:
2005:
2002:
1999:
1996:
1993:
1990:
1966:
1961:
1957:
1953:
1948:
1944:
1940:
1935:
1931:
1910:
1907:
1904:
1901:
1898:
1895:
1892:
1880:
1877:
1876:
1875:
1864:
1853:
1842:
1837:
1831:
1822:
1816:
1811:for the lines
1805:
1800:
1794:
1785:
1779:
1768:
1759:
1752:
1745:
1738:
1731:
1724:
1717:
1685:
1676:
1663:
1652:
1637:
1626:
1617:
1616:
1607:of the circle
1603:
1594:
1587:
1533:
1532:
1517:
1508:is drawn from
1485:
1482:
1466:
1463:
1460:
1458:
1456:
1453:
1450:
1445:
1441:
1437:
1434:
1429:
1425:
1421:
1418:
1415:
1412:
1409:
1406:
1403:
1398:
1394:
1390:
1387:
1382:
1378:
1374:
1371:
1368:
1365:
1364:
1361:
1358:
1355:
1350:
1346:
1342:
1339:
1334:
1330:
1326:
1323:
1320:
1317:
1314:
1311:
1309:
1307:
1304:
1301:
1296:
1292:
1288:
1285:
1280:
1276:
1272:
1269:
1266:
1263:
1262:
1256:
1253:
1248:
1244:
1238:
1235:
1230:
1226:
1219:
1216:
1213:
1208:
1204:
1200:
1197:
1194:
1192:
1188:
1184:
1180:
1177:
1174:
1173:
1167:
1164:
1159:
1155:
1149:
1146:
1141:
1137:
1128:
1124:
1120:
1115:
1111:
1107:
1104:
1098:
1095:
1090:
1086:
1080:
1077:
1072:
1068:
1061:
1058:
1055:
1053:
1051:
1048:
1047:
1002:
999:
994:
990:
984:
981:
976:
972:
965:
962:
959:
957:
952:
949:
944:
941:
935:
934:
931:
928:
925:
923:
918:
915:
910:
907:
901:
898:
895:
892:
889:
886:
883:
880:
877:
874:
871:
868:
865:
862:
861:
856:
852:
848:
845:
843:
839:
835:
831:
828:
825:
822:
819:
814:
810:
806:
803:
800:
794:
789:
788:
762:
757:
753:
749:
744:
740:
736:
733:
728:
724:
720:
717:
712:
708:
704:
701:
696:
692:
688:
674:
667:
650:
645:
641:
637:
632:
628:
624:
621:
618:
615:
612:
607:
603:
599:
596:
593:
590:
564:
561:
558:
555:
552:
547:
543:
539:
536:
531:
527:
523:
520:
517:
514:
511:
508:
505:
500:
496:
492:
489:
484:
480:
476:
473:
470:
454:
447:
416:
412:
408:
403:
399:
395:
392:
389:
386:
383:
378:
374:
370:
367:
364:
361:
345:
342:
219:in the circle
98:
95:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
8332:
8321:
8318:
8317:
8315:
8303:
8302:
8297:
8294:
8289:
8284:
8283:
8278:
8275:
8270:
8269:
8265:
8258:
8254:
8250:
8246:
8242:
8238:
8231:
8224:
8221:
8217:
8214:, p. 110, at
8213:
8212:
8205:
8198:
8195:
8183:
8180:Paul Kunkel.
8176:
8173:
8170:
8167:
8162:
8159:
8154:
8153:
8148:
8142:
8139:
8133:
8131:
8118:
8111:
8108:
8103:
8100:
8093:
8087:
8081:
8078:
8068:
8063:
8058:
8054:
8031:
8028:
8023:
8020:
8004:
8000:
7979:
7973:
7970:
7967:
7964:
7961:
7958:
7955:
7949:
7942:
7939:
7934:
7931:
7917:
7913:
7906:
7902:
7897:
7891:
7887:
7883:
7879:
7874:
7870:
7865:
7863:
7859:
7855:
7851:
7846:
7842:
7840:
7836:
7832:
7828:
7822:
7818:
7810:
7789:
7782:
7780:
7777:
7773:
7769:
7764:
7761:
7757:
7747:
7742:
7734:
7732:
7730:
7729:Gaspard Monge
7726:
7718:
7712:
7703:
7697:
7688:
7682:
7673:
7664:
7655:
7648:
7638:
7636:
7632:
7631:
7624:
7616:
7611:
7609:
7605:
7601:
7586:
7581:
7573:
7570:
7564:
7559:
7555:
7551:
7546:
7542:
7532:
7507:
7503:
7499:
7494:
7490:
7481:
7478:
7470:
7448:
7444:
7440:
7435:
7431:
7422:
7419:
7411:
7394:
7390:
7386:
7381:
7377:
7373:
7370:
7367:
7357:
7353:
7349:
7344:
7340:
7326:
7309:
7305:
7301:
7296:
7292:
7288:
7285:
7277:
7274:
7256:
7252:
7248:
7243:
7239:
7235:
7232:
7224:
7223:
7222:
7219:
7215:
7206:
7204:
7201:
7187:
7182:
7178:
7174:
7169:
7165:
7155:
7136:
7131:
7127:
7123:
7121:
7111:
7107:
7103:
7098:
7094:
7087:
7079:
7075:
7071:
7066:
7062:
7049:
7044:
7040:
7036:
7034:
7024:
7020:
7016:
7011:
7007:
7000:
6992:
6988:
6984:
6979:
6975:
6964:
6961:
6959:
6949:
6945:
6941:
6936:
6932:
6925:
6917:
6913:
6909:
6904:
6900:
6889:
6886:
6884:
6874:
6870:
6866:
6861:
6857:
6850:
6842:
6838:
6834:
6829:
6825:
6809:
6803:
6794:
6785:
6776:
6767:
6758:
6748:
6741:
6739:
6736:
6732:
6725:
6716:
6707:
6698:
6689:
6676:
6667:
6660:
6656:
6649:
6640:
6633:
6629:
6628:
6627:
6612:
6606:
6597:
6593:
6589:
6586:
6581:
6570:
6566:
6562:
6559:
6554:
6549:
6543:
6532:
6524:
6520:
6494:
6490:
6486:
6483:
6478:
6469:
6458:
6444:
6435:
6428:
6424:
6418:
6394:
6390:
6386:
6383:
6378:
6374:
6370:
6364:
6359:
6355:
6351:
6349:
6344:
6333:
6329:
6325:
6322:
6317:
6314:
6311:
6308:
6305:
6302:
6300:
6295:
6284:
6280:
6276:
6273:
6268:
6265:
6262:
6259:
6256:
6253:
6251:
6246:
6232:
6211:
6208:
6205:
6203:
6196:
6192:
6188:
6183:
6179:
6171:
6168:
6165:
6163:
6158:
6155:
6152:
6149:
6146:
6120:
6116:
6107:
6104:
6100:
6095:
6091:
6082:
6079:
6075:
6070:
6066:
6057:
6054:
6042:
6033:
6012:
6004:
5995:
5990:
5982:
5971:
5968:
5959:
5943:
5939:
5935:
5930:
5926:
5922:
5919:
5894:
5890:
5886:
5881:
5877:
5873:
5870:
5845:
5841:
5837:
5832:
5828:
5824:
5821:
5814:
5809:
5805:
5801:
5796:
5792:
5788:
5785:
5759:
5752:
5749:
5743:
5740:
5734:
5731:
5728:
5722:
5712:
5708:
5704:
5680:
5676:
5672:
5670:
5665:
5662:
5657:
5653:
5649:
5646:
5641:
5637:
5633:
5624:
5620:
5616:
5614:
5609:
5606:
5601:
5597:
5593:
5590:
5585:
5581:
5577:
5553:
5550:
5547:
5542:
5538:
5534:
5529:
5525:
5504:
5501:
5498:
5495:
5492:
5489:
5486:
5483:
5480:
5477:
5465:
5456:
5446:
5439:
5432:
5422:
5415:
5408:
5399:
5397:
5391:
5382:
5373:
5364:
5355:
5346:
5337:
5328:
5319:
5312:
5303:
5299:A new circle
5293:
5286:
5285:
5284:
5278:
5269:
5260:
5251:
5242:
5233:
5224:
5215:
5206:
5197:
5190:
5181:
5177:A new circle
5171:
5164:
5163:
5162:
5156:
5147:
5140:
5135:
5127:
5118:
5109:
5100:
5091:
5082:
5073:
5071:
5069:
5060:
5058:
5051:
5045:Inner tangent
5044:
5042:
5036:
5030:
5017:
5008:
4999:
4990:
4964:
4960:
4956:
4951:
4947:
4941:
4938:
4923:
4915:
4906:
4896:
4892:
4890:
4869:
4861:
4851:
4847:
4843:
4838:
4834:
4827:
4822:
4812:
4808:
4804:
4799:
4795:
4786:
4783:
4780:
4774:
4770:
4767:
4764:
4761:
4759:
4754:
4746:
4739:
4735:
4731:
4726:
4722:
4717:
4711:
4707:
4703:
4698:
4694:
4688:
4679:
4676:
4674:
4669:
4656:
4652:
4648:
4630:
4613:
4610:
4607:
4604:
4601:
4596:
4592:
4588:
4586:
4579:
4575:
4567:
4564:
4561:
4558:
4555:
4550:
4546:
4542:
4540:
4533:
4529:
4521:
4518:
4515:
4512:
4509:
4504:
4500:
4496:
4494:
4487:
4483:
4475:
4472:
4469:
4466:
4463:
4458:
4454:
4450:
4448:
4441:
4437:
4424:
4413:
4406:
4395:
4388:
4377:
4370:
4359:
4352:
4341:
4334:
4323:
4316:
4305:
4299:Outer tangent
4298:
4296:
4294:
4290:
4286:
4282:
4277:
4273:
4262:
4257:
4250:
4248:
4246:
4245:parallelogram
4242:
4237:
4234:
4231:
4211:
4208:
4205:
4199:
4193:
4190:
4187:
4181:
4172:
4169:
4163:
4154:
4151:
4141:
4131:
4128:
4125:
4119:
4113:
4110:
4107:
4101:
4092:
4089:
4083:
4074:
4071:
4055:
4037:
4034:
4031:
4022:
4019:
4013:
4004:
4001:
3992:
3989:
3986:
3977:
3974:
3968:
3959:
3956:
3946:
3943:
3940:
3931:
3928:
3922:
3913:
3910:
3901:
3898:
3895:
3886:
3883:
3877:
3868:
3865:
3801:
3797:
3784:
3775:
3772:
3766:
3757:
3754:
3748:
3739:
3736:
3730:
3721:
3718:
3707:
3705:
3704:Pitot theorem
3697:
3682:
3674:
3672:
3670:
3666:
3662:
3658:
3654:
3650:
3642:
3640:
3627:
3622:
3618:
3614:
3609:
3605:
3601:
3594:has equation
3578:
3575:
3570:
3566:
3554:
3552:
3546:
3533:
3527:
3524:
3519:
3515:
3489:
3485:
3481:
3476:
3472:
3468:
3460:
3459:
3439:
3434:
3430:
3426:
3423:
3418:
3414:
3410:
3407:
3402:
3398:
3391:
3386:
3382:
3378:
3375:
3370:
3366:
3362:
3359:
3354:
3350:
3340:
3333:
3328:
3314:
3309:
3305:
3301:
3298:
3293:
3289:
3285:
3282:
3277:
3273:
3260:
3254:
3247:
3242:
3238:
3231:
3226:
3225:
3224:
3211:
3198:
3194:
3187:
3183:
3179:
3163:
3159:
3155:
3150:
3145:
3141:
3131:
3126:
3122:
3118:
3113:
3100:
3096:
3090:
3086:
3070:
3065:
3061:
3055:
3051:
3045:
3032:
3028:
3024:
3020:
3014:
3010:
3006:
3002:
2981:
2964:
2951:
2947:
2940:
2936:
2932:
2916:
2912:
2908:
2903:
2898:
2888:
2878:
2865:
2861:
2855:
2851:
2835:
2831:
2827:
2822:
2817:
2807:
2779:
2774:
2770:
2766:
2761:
2756:
2752:
2746:
2741:
2737:
2724:
2698:
2694:
2690:
2685:
2681:
2674:
2669:
2665:
2655:
2642:
2628:
2625:
2610:
2606:
2602:
2597:
2592:
2588:
2578:
2574:
2570:
2565:
2554:
2551:
2536:
2532:
2526:
2522:
2516:
2503:
2499:
2495:
2467:
2460:
2456:
2452:
2447:
2442:
2438:
2428:
2424:
2420:
2415:
2412:
2405:
2401:
2397:
2392:
2388:
2382:
2379:
2374:
2370:
2365:
2358:
2354:
2348:
2344:
2338:
2335:
2310:
2306:
2302:
2299:
2293:
2288:
2284:
2280:
2276:
2263:
2246:
2240:
2237:
2232:
2228:
2221:
2216:
2212:
2198:
2194:
2181:
2176:
2172:
2168:
2163:
2159:
2155:
2132:
2129:
2124:
2120:
2113:
2108:
2104:
2077:
2069:
2066:
2063:
2057:
2047:
2044:
2016:
2011:
2007:
2003:
2000:
1997:
1994:
1991:
1988:
1981:has equation
1964:
1959:
1955:
1951:
1946:
1942:
1938:
1933:
1929:
1905:
1902:
1899:
1893:
1890:
1878:
1865:
1854:
1843:
1836:
1830:
1821:
1815:
1806:
1799:
1793:
1784:
1778:
1769:
1758:
1751:
1744:
1737:
1730:
1723:
1718:
1711:
1710:
1709:
1707:
1699:
1694:
1684:
1675:
1661:
1650:
1635:
1624:
1602:
1593:
1588:
1565:
1564:
1563:
1553:
1549:
1537:
1526:
1525:perpendicular
1518:
1503:
1502:
1501:
1491:
1483:
1481:
1464:
1461:
1459:
1451:
1448:
1443:
1439:
1427:
1423:
1419:
1416:
1410:
1404:
1401:
1396:
1392:
1380:
1376:
1372:
1369:
1356:
1353:
1348:
1344:
1332:
1328:
1324:
1321:
1315:
1312:
1310:
1302:
1299:
1294:
1290:
1278:
1274:
1270:
1267:
1254:
1251:
1246:
1242:
1236:
1233:
1228:
1224:
1214:
1211:
1206:
1202:
1195:
1193:
1186:
1182:
1178:
1175:
1165:
1162:
1157:
1153:
1147:
1144:
1139:
1135:
1126:
1122:
1118:
1113:
1109:
1105:
1102:
1096:
1093:
1088:
1084:
1078:
1075:
1070:
1066:
1059:
1056:
1054:
1049:
1037:
1034:
1030:
1026:
1020:
1000:
997:
992:
988:
982:
979:
974:
970:
963:
960:
958:
950:
947:
942:
939:
929:
926:
924:
916:
913:
908:
905:
896:
893:
890:
884:
881:
875:
872:
869:
863:
854:
850:
846:
844:
837:
829:
826:
823:
817:
812:
804:
801:
798:
778:
760:
755:
751:
747:
742:
734:
731:
726:
722:
715:
710:
702:
699:
694:
690:
673:
666:
648:
643:
639:
635:
630:
622:
619:
616:
610:
605:
597:
594:
591:
580:
575:
562:
559:
553:
550:
545:
541:
529:
525:
521:
518:
512:
506:
503:
498:
494:
482:
478:
474:
471:
460:
453:
446:
438:
434:
414:
410:
406:
401:
393:
390:
387:
381:
376:
368:
365:
362:
351:
343:
341:
338:
334:
326:
309:
304:
302:
301:supplementary
297:
290:
263:
257:
236:
232:
218:
214:
194:
166:
161:
157:
155:
149:
147:
143:
139:
135:
131:
127:
123:
119:
111:
96:
94:
92:
88:
84:
80:
79:perpendicular
73:
69:
65:
62:. Since the
61:
57:
56:constructions
53:
49:
45:
41:
37:
33:
19:
8299:
8280:
8243:(1): 34–46,
8240:
8236:
8223:
8216:Google Books
8209:
8203:
8197:
8186:. Retrieved
8175:
8161:
8150:
8141:
8066:
8056:
8052:
8002:
7998:
7915:
7911:
7904:
7900:
7889:
7885:
7881:
7877:
7866:
7847:
7843:
7833:) case of a
7824:
7767:
7765:
7759:
7753:
7727:altogether.
7716:
7710:
7701:
7695:
7686:
7680:
7671:
7662:
7653:
7650:
7634:
7630:belt problem
7628:
7626:
7623:Belt problem
7617:Belt problem
7612:Applications
7606:
7602:
7533:
7530:
7271:), which is
7210:
7202:
7156:
6810:
6801:
6792:
6783:
6774:
6765:
6756:
6753:
6737:
6730:
6723:
6714:
6705:
6696:
6687:
6684:
6674:
6665:
6658:
6647:
6638:
6631:
6530:
6522:
6518:
6467:
6456:
6442:
6433:
6426:
6422:
6419:
6230:
6040:
6031:
5960:
5710:
5706:
5702:
5699:Solving for
5463:
5454:
5452:with radius
5444:
5437:
5430:
5420:
5413:
5406:
5403:
5389:
5380:
5371:
5362:
5353:
5344:
5335:
5326:
5317:
5310:
5301:
5298:
5276:
5267:
5258:
5249:
5240:
5231:
5222:
5213:
5204:
5195:
5188:
5179:
5176:
5154:
5145:
5138:
5125:
5116:
5107:
5098:
5089:
5080:
5077:
5064:
5061:Construction
5056:
5034:
5028:
5015:
5006:
4997:
4988:
4913:
4904:
4901:
4654:
4650:
4646:
4631:
4425:
4411:
4404:
4393:
4386:
4375:
4368:
4357:
4350:
4339:
4332:
4321:
4314:
4310:
4285:affine plane
4269:
4238:
4235:
4232:
4056:
3841:(on segment
3831:(on segment
3821:(on segment
3811:(on segment
3806:
3708:
3678:
3646:
3549:Relation to
3548:
3547:
3456:Relation to
3455:
3454:
3338:
3331:
3258:
3252:
3245:
3236:
3229:
2979:
2722:
2656:
2261:
2203:
1882:
1834:
1828:
1819:
1813:
1797:
1791:
1782:
1776:
1756:
1749:
1742:
1735:
1728:
1721:
1706:straightedge
1695:
1682:
1673:
1659:
1648:
1633:
1622:
1618:
1600:
1591:
1546:
1487:
1038:
1032:
1028:
1024:
1021:
779:
671:
664:
576:
461:
451:
444:
436:
432:
347:
336:
332:
324:
305:
295:
288:
264:
245:
192:
190:
177:for any ray
150:
134:translations
118:secant lines
105:to a circle
100:
87:radial lines
64:tangent line
35:
29:
8211:online copy
4383:the points
3851:are equal:
3461:: Equation
3329:In case of
265:If a point
8188:2008-09-29
8134:References
7896:derivative
7884:) = (cosh
7839:bitangents
6790:and radii
6733:= −1
6661:= −1
5308:of radius
5186:of radius
4247:does not.
3702:. By the
1807:Similarly
1804:intersect,
238:The angle
138:inversions
124:, such as
110:intersects
91:orthogonal
8301:MathWorld
8282:MathWorld
8257:122408307
7971:
7959:
7920:, and is
7831:reducible
7776:inversion
7571:−
7500:−
7441:−
7350:−
7175:−
7104:−
7088:⋅
7072:−
7017:−
7001:⋅
6985:−
6942:−
6926:⋅
6910:−
6867:−
6851:⋅
6835:−
6590:−
6582:±
6563:−
6555:∓
6487:−
6479:±
6365:−
6326:−
6277:−
6263:−
6114:Δ
6089:Δ
6064:Δ
6002:Δ
5980:Δ
5917:Δ
5887:−
5868:Δ
5838:−
5819:Δ
5802:−
5783:Δ
5747:Δ
5738:Δ
5726:Δ
5132:be their
4957:−
4844:−
4805:−
4784:−
4771:
4765:±
4755:β
4732:−
4704:−
4680:−
4670:γ
4614:α
4611:
4602:±
4568:α
4565:
4556:±
4522:α
4519:
4510:±
4476:α
4473:
4464:±
4287:; in the
4272:bitangent
4177:¯
4159:¯
4097:¯
4079:¯
4027:¯
4009:¯
3982:¯
3964:¯
3936:¯
3918:¯
3891:¯
3873:¯
3780:¯
3762:¯
3744:¯
3726:¯
3696:inscribed
3669:congruent
3180:−
3156:−
3114:±
2933:−
2892:→
2811:→
2657:If point
2603:−
2566:±
2504:±
2453:−
2416:±
2398:−
2383:±
2375:±
2311:±
2052:→
1698:construct
1552:construct
1519:The line
1490:construct
1449:−
1420:−
1402:−
1373:−
1354:−
1325:−
1316:−
1300:−
1271:−
1252:−
1234:−
1212:−
1179:−
1163:−
1145:−
1094:−
1076:−
1060:−
998:−
980:−
964:−
894:−
873:−
827:−
802:−
732:−
700:−
620:−
595:−
551:−
522:−
504:−
475:−
391:−
366:−
93:circles.
8314:Category
7809:inverses
7216:and the
6471:is then
6449:, while
5194:−
5114:and let
4653:−
3661:triangle
3659:. Every
3657:incircle
2029:because
1578:, where
1527:line to
130:rotation
126:scalings
52:theorems
8320:Circles
8064:(where
8048:
8009:
7888:, sinh
6742:Vectors
5775:where
5136:, with
4986:can be
3653:polygon
1523:is the
1504:A line
1492:a line
285:, then
163:By the
81:to the
8255:
7996:since
7894:. The
7850:Möbius
7708:, and
7669:, and
6460:where
4982:Here,
4887:where
4768:arcsin
4658:with
4655:β
4651:γ
4647:α
4642:α
4421:α
4241:square
3837:) and
3395:
3256:point
2882:
2640:
679:where
140:, and
83:radius
68:circle
60:proofs
44:circle
8253:S2CID
8233:(PDF)
7772:scale
5144:>
5134:radii
4889:atan2
4684:atan2
4632:Here
3651:is a
3337:>
3235:<
2328:with
2148:with
335:= ½ ∠
329:then
327:≤ 90°
308:chord
306:If a
72:point
70:at a
66:to a
48:point
38:is a
8069:= +1
8007:and
7968:cosh
7956:sinh
7819:and
7807:are
7803:and
7482:<
7374:<
7368:<
7236:>
6799:and
6781:and
6763:and
6728:and
6466:sin
6455:cos
6233:= ±1
5860:and
5461:and
5428:and
5229:and
5123:and
5105:and
5087:and
5078:Let
5013:and
4911:and
4636:and
4329:and
3849:ABCD
3700:ABCD
3684:ABCD
3243:For
3227:For
1883:Let
1870:and
1859:and
1848:and
1826:and
1789:and
1770:Let
1719:Let
1680:and
1656:and
1630:and
1598:and
321:and
299:are
292:and
281:and
89:and
58:and
40:line
34:, a
8245:doi
8067:j j
7898:of
7852:or
7848:In
7768:LLC
7760:LLL
6726:= 1
6672:to
6645:to
6634:= 1
6527:by
6453:is
6440:to
6038:to
5961:If
5436:= (
5412:= (
4995:or
4924:),
4608:cos
4562:sin
4516:cos
4470:sin
3827:),
3817:),
3694:is
352:is
337:TOM
333:PTM
325:PTM
296:TOS
289:TPS
193:two
179:PMN
77:is
30:In
8316::
8298:.
8279:.
8251:,
8241:22
8239:,
8235:,
8149:.
8055:=
7864:.
7693:,
7660:,
7637:.
6735:.
6521:,
6425:,
5709:,
5705:,
5443:,
5419:,
5316:+
5022:.
4649:=
4423::
4410:,
4401:,
4392:,
4374:,
4365:,
4356:,
4338:,
4320:,
3844:DA
3834:CD
3824:BC
3814:AB
3679:A
3647:A
3251:=
1872:PG
1868:PF
1755:,
1748:,
1741:,
1734:,
1727:,
1708::
1693:.
1660:PT
1649:PT
1634:OT
1623:OT
1575:OP
1569:OP
1562::
1543:).
1036:.
1031:+
1029:kx
1027:=
777::
670:,
450:,
435:,
340:.
311:TM
231:.
184:PT
174:PN
172:·
170:PM
136:,
132:,
128:,
8304:.
8285:.
8247::
8218:)
8208:(
8191:.
8119:.
8112:a
8109:d
8104:p
8101:d
8094:=
8091:)
8088:a
8085:(
8082:p
8079:j
8057:x
8053:y
8032:a
8029:d
8024:p
8021:d
8005:)
8003:a
8001:(
7999:p
7994:a
7980:.
7977:)
7974:a
7965:,
7962:a
7953:(
7950:=
7943:a
7940:d
7935:p
7932:d
7918:)
7916:a
7914:(
7912:p
7907:)
7905:a
7903:(
7901:p
7892:)
7890:a
7886:a
7882:a
7880:(
7878:p
7805:Q
7801:P
7797:q
7793:Q
7720:3
7717:C
7714:1
7711:C
7705:3
7702:C
7699:2
7696:C
7690:2
7687:C
7684:1
7681:C
7675:3
7672:C
7666:2
7663:C
7657:1
7654:C
7587:,
7582:2
7578:)
7574:r
7568:(
7565:=
7560:2
7556:y
7552:+
7547:2
7543:x
7514:|
7508:2
7504:r
7495:1
7491:r
7486:|
7479:d
7455:|
7449:2
7445:r
7436:1
7432:r
7427:|
7423:=
7420:d
7395:2
7391:r
7387:+
7382:1
7378:r
7371:d
7364:|
7358:2
7354:r
7345:1
7341:r
7336:|
7310:2
7306:r
7302:+
7297:1
7293:r
7289:=
7286:d
7257:2
7253:r
7249:+
7244:1
7240:r
7233:d
7188:,
7183:1
7179:t
7170:2
7166:t
7137:2
7132:2
7128:r
7124:=
7117:)
7112:2
7108:v
7099:2
7095:t
7091:(
7085:)
7080:2
7076:v
7067:2
7063:t
7059:(
7050:2
7045:1
7041:r
7037:=
7030:)
7025:1
7021:v
7012:1
7008:t
7004:(
6998:)
6993:1
6989:v
6980:1
6976:t
6972:(
6965:0
6962:=
6955:)
6950:1
6946:t
6937:2
6933:t
6929:(
6923:)
6918:1
6914:v
6905:1
6901:t
6897:(
6890:0
6887:=
6880:)
6875:1
6871:t
6862:2
6858:t
6854:(
6848:)
6843:2
6839:v
6830:2
6826:t
6822:(
6805:2
6802:r
6796:1
6793:r
6787:2
6784:v
6778:1
6775:v
6769:2
6766:t
6760:1
6757:t
6731:k
6724:k
6718:2
6715:c
6709:1
6706:c
6700:2
6697:r
6691:1
6688:r
6681:.
6678:1
6675:c
6669:2
6666:c
6659:k
6654:.
6651:2
6648:c
6642:1
6639:c
6632:k
6613:)
6607:R
6598:2
6594:R
6587:1
6571:2
6567:R
6560:1
6550:R
6544:(
6531:θ
6529:±
6525:)
6523:Y
6519:X
6517:(
6513:θ
6495:2
6491:R
6484:1
6468:θ
6462:θ
6457:θ
6451:R
6446:2
6443:c
6437:1
6434:c
6429:)
6427:Y
6423:X
6421:(
6400:)
6395:1
6391:y
6387:b
6384:+
6379:1
6375:x
6371:a
6368:(
6360:1
6356:r
6352:=
6345:c
6334:2
6330:R
6323:1
6318:X
6315:k
6312:+
6309:Y
6306:R
6303:=
6296:b
6285:2
6281:R
6274:1
6269:Y
6266:k
6260:X
6257:R
6254:=
6247:a
6231:k
6212:;
6209:1
6206:=
6197:2
6193:b
6189:+
6184:2
6180:a
6172:,
6169:R
6166:=
6159:Y
6156:b
6153:+
6150:X
6147:a
6121:d
6117:r
6108:=
6105:R
6101:,
6096:d
6092:y
6083:=
6080:Y
6076:,
6071:d
6067:x
6058:=
6055:X
6044:2
6041:c
6035:1
6032:c
6013:2
6009:)
6005:y
5999:(
5996:+
5991:2
5987:)
5983:x
5977:(
5972:=
5969:d
5944:1
5940:r
5936:+
5931:2
5927:r
5923:=
5920:r
5895:1
5891:r
5882:2
5878:r
5874:=
5871:r
5846:1
5842:y
5833:2
5829:y
5825:=
5822:y
5815:,
5810:1
5806:x
5797:2
5793:x
5789:=
5786:x
5763:)
5760:1
5757:(
5753:,
5750:r
5744:=
5741:y
5735:b
5732:+
5729:x
5723:a
5713:)
5711:c
5707:b
5703:a
5701:(
5681:2
5677:r
5673:=
5666:c
5663:+
5658:2
5654:y
5650:b
5647:+
5642:2
5638:x
5634:a
5625:1
5621:r
5617:=
5610:c
5607:+
5602:1
5598:y
5594:b
5591:+
5586:1
5582:x
5578:a
5554:,
5551:1
5548:=
5543:2
5539:b
5535:+
5530:2
5526:a
5505:,
5502:0
5499:=
5496:c
5493:+
5490:y
5487:b
5484:+
5481:x
5478:a
5467:2
5464:r
5458:1
5455:r
5450:)
5448:2
5445:y
5441:2
5438:x
5434:2
5431:c
5426:)
5424:1
5421:y
5417:1
5414:x
5410:1
5407:c
5393:1
5390:C
5384:3
5381:C
5375:1
5372:O
5366:2
5363:r
5357:1
5354:C
5348:2
5345:C
5339:2
5336:O
5330:1
5327:O
5321:2
5318:r
5314:1
5311:r
5305:3
5302:C
5280:1
5277:C
5271:3
5268:C
5262:1
5259:O
5253:2
5250:C
5244:2
5241:r
5235:2
5232:C
5226:1
5223:C
5217:2
5214:O
5208:1
5205:O
5199:2
5196:r
5192:1
5189:r
5183:3
5180:C
5158:1
5155:C
5149:2
5146:r
5142:1
5139:r
5129:2
5126:r
5120:1
5117:r
5111:2
5108:C
5102:1
5099:C
5093:2
5090:O
5084:1
5081:O
5038:2
5035:O
5032:1
5029:O
5024:d
5019:1
5016:O
5010:2
5007:O
5001:2
4998:r
4992:1
4989:r
4984:r
4965:2
4961:r
4952:1
4948:r
4942:r
4939:d
4926:S
4917:1
4914:O
4908:2
4905:O
4870:)
4862:2
4858:)
4852:1
4848:y
4839:2
4835:y
4831:(
4828:+
4823:2
4819:)
4813:1
4809:x
4800:2
4796:x
4792:(
4787:r
4781:R
4775:(
4762:=
4747:)
4740:1
4736:x
4727:2
4723:x
4718:,
4712:1
4708:y
4699:2
4695:y
4689:(
4677:=
4638:r
4634:R
4605:R
4597:2
4593:y
4589:=
4580:4
4576:y
4559:R
4551:2
4547:x
4543:=
4534:4
4530:x
4513:r
4505:1
4501:y
4497:=
4488:3
4484:y
4467:r
4459:1
4455:x
4451:=
4442:3
4438:x
4417:)
4415:4
4412:y
4408:4
4405:x
4403:(
4399:)
4397:3
4394:y
4390:3
4387:x
4385:(
4381:)
4379:2
4376:y
4372:2
4369:x
4367:(
4363:)
4361:1
4358:y
4354:1
4351:x
4349:(
4345:)
4343:4
4340:y
4336:4
4333:x
4331:(
4327:)
4325:3
4322:y
4318:3
4315:x
4313:(
4264:S
4215:)
4212:a
4209:+
4206:d
4203:(
4200:+
4197:)
4194:c
4191:+
4188:b
4185:(
4182:=
4173:A
4170:D
4164:+
4155:C
4152:B
4142:=
4135:)
4132:d
4129:+
4126:c
4123:(
4120:+
4117:)
4114:b
4111:+
4108:a
4105:(
4102:=
4093:D
4090:C
4084:+
4075:B
4072:A
4038:.
4035:a
4032:=
4023:P
4020:A
4014:=
4005:S
4002:A
3993:,
3990:d
3987:=
3978:S
3975:D
3969:=
3960:R
3957:D
3947:,
3944:c
3941:=
3932:R
3929:C
3923:=
3914:Q
3911:C
3902:,
3899:b
3896:=
3887:Q
3884:B
3878:=
3869:P
3866:B
3839:S
3829:R
3819:Q
3809:P
3785:.
3776:A
3773:D
3767:+
3758:C
3755:B
3749:=
3740:D
3737:C
3731:+
3722:B
3719:A
3692:C
3688:C
3628:.
3623:2
3619:r
3615:=
3610:0
3606:x
3602:x
3582:)
3579:0
3576:,
3571:0
3567:x
3563:(
3553::
3534:.
3531:)
3528:0
3525:,
3520:0
3516:x
3512:(
3490:2
3486:r
3482:=
3477:0
3473:x
3469:a
3440:.
3435:2
3431:r
3427:=
3424:y
3419:2
3415:y
3411:+
3408:x
3403:2
3399:x
3392:,
3387:2
3383:r
3379:=
3376:y
3371:1
3367:y
3363:+
3360:x
3355:1
3351:x
3339:r
3335:0
3332:d
3315:.
3310:2
3306:r
3302:=
3299:y
3294:0
3290:y
3286:+
3283:x
3278:0
3274:x
3262:0
3259:P
3253:r
3249:0
3246:d
3237:r
3233:0
3230:d
3212:.
3206:)
3199:0
3195:x
3188:0
3184:y
3174:(
3164:2
3160:r
3151:2
3146:0
3142:d
3132:2
3127:0
3123:d
3119:r
3108:)
3101:0
3097:y
3091:0
3087:x
3081:(
3071:2
3066:0
3062:d
3056:2
3052:r
3046:=
3040:)
3033:2
3029:/
3025:1
3021:y
3015:2
3011:/
3007:1
3003:x
2997:(
2983:0
2980:P
2965:.
2959:)
2952:0
2948:x
2941:0
2937:y
2927:(
2917:0
2913:d
2909:1
2904:=
2899:2
2889:e
2879:,
2873:)
2866:0
2862:y
2856:0
2852:x
2846:(
2836:0
2832:d
2828:1
2823:=
2818:1
2808:e
2780:2
2775:0
2771:y
2767:+
2762:2
2757:0
2753:x
2747:=
2742:0
2738:d
2726:0
2723:x
2718:x
2704:)
2699:0
2695:y
2691:,
2686:0
2682:x
2678:(
2675:=
2670:0
2666:P
2643:.
2634:)
2629:1
2626:0
2621:(
2611:2
2607:r
2598:2
2593:0
2589:x
2579:0
2575:x
2571:r
2560:)
2555:0
2552:1
2547:(
2537:0
2533:x
2527:2
2523:r
2517:=
2511:)
2500:b
2496:a
2491:(
2468:.
2461:2
2457:r
2448:2
2443:0
2439:x
2429:0
2425:x
2421:r
2413:=
2406:2
2402:a
2393:2
2389:r
2380:=
2371:b
2366:,
2359:0
2355:x
2349:2
2345:r
2339:=
2336:a
2316:)
2307:b
2303:,
2300:a
2297:(
2294:=
2289:2
2285:/
2281:1
2277:P
2265:0
2262:P
2247:,
2244:)
2241:0
2238:,
2233:0
2229:x
2225:(
2222:=
2217:0
2213:P
2182:.
2177:2
2173:r
2169:=
2164:0
2160:x
2156:a
2136:)
2133:0
2130:,
2125:0
2121:x
2117:(
2114:=
2109:0
2105:P
2094:x
2078:T
2074:)
2070:b
2067:,
2064:a
2061:(
2058:=
2048:P
2045:O
2031:P
2017:,
2012:2
2008:r
2004:=
2001:y
1998:b
1995:+
1992:x
1989:a
1979:P
1965:.
1960:2
1956:r
1952:=
1947:2
1943:y
1939:+
1934:2
1930:x
1909:)
1906:b
1903:,
1900:a
1897:(
1894:=
1891:P
1874:.
1863:.
1861:G
1857:F
1852:.
1850:E
1846:D
1841:.
1838:1
1835:C
1832:2
1829:B
1823:2
1820:C
1817:1
1814:B
1809:E
1801:1
1798:B
1795:2
1792:A
1786:2
1783:B
1780:1
1777:A
1772:D
1767:.
1765:P
1760:2
1757:C
1753:1
1750:C
1746:2
1743:B
1739:1
1736:B
1732:2
1729:A
1725:1
1722:A
1714:P
1702:P
1691:C
1686:2
1683:T
1677:1
1674:T
1669:P
1664:2
1653:1
1643:C
1638:2
1627:1
1613:P
1609:C
1604:2
1601:T
1595:1
1592:T
1586:.
1584:C
1580:O
1560:C
1556:P
1541:P
1531:.
1529:a
1521:t
1516:;
1514:T
1510:O
1506:a
1498:T
1494:t
1465:0
1462:=
1455:)
1452:b
1444:1
1440:y
1436:(
1433:)
1428:1
1424:y
1417:y
1414:(
1411:+
1408:)
1405:a
1397:1
1393:x
1389:(
1386:)
1381:1
1377:x
1370:x
1367:(
1360:)
1357:a
1349:1
1345:x
1341:(
1338:)
1333:1
1329:x
1322:x
1319:(
1313:=
1306:)
1303:b
1295:1
1291:y
1287:(
1284:)
1279:1
1275:y
1268:y
1265:(
1255:b
1247:1
1243:y
1237:a
1229:1
1225:x
1218:)
1215:x
1207:1
1203:x
1199:(
1196:=
1187:1
1183:y
1176:y
1166:b
1158:1
1154:y
1148:a
1140:1
1136:x
1127:1
1123:x
1119:+
1114:1
1110:y
1106:+
1103:x
1097:b
1089:1
1085:y
1079:a
1071:1
1067:x
1057:=
1050:y
1033:m
1025:y
1001:b
993:1
989:y
983:a
975:1
971:x
961:=
951:x
948:d
943:y
940:d
930:0
927:=
917:x
914:d
909:y
906:d
900:)
897:b
891:y
888:(
885:2
882:+
879:)
876:a
870:x
867:(
864:2
855:2
851:r
847:=
838:2
834:)
830:b
824:y
821:(
818:+
813:2
809:)
805:a
799:x
793:(
775:x
761:.
756:2
752:r
748:=
743:2
739:)
735:b
727:1
723:y
719:(
716:+
711:2
707:)
703:a
695:1
691:x
687:(
677:)
675:1
672:y
668:1
665:x
663:(
649:,
644:2
640:r
636:=
631:2
627:)
623:b
617:y
614:(
611:+
606:2
602:)
598:a
592:x
589:(
563:0
560:=
557:)
554:b
546:1
542:y
538:(
535:)
530:1
526:y
519:y
516:(
513:+
510:)
507:a
499:1
495:x
491:(
488:)
483:1
479:x
472:x
469:(
457:)
455:1
452:y
448:1
445:x
443:(
439:)
437:b
433:a
431:(
415:2
411:r
407:=
402:2
398:)
394:b
388:y
385:(
382:+
377:2
373:)
369:a
363:x
360:(
331:∠
323:∠
319:P
315:T
294:∠
287:∠
283:S
279:T
275:P
271:O
267:P
260:P
252:T
248:t
240:θ
229:P
225:P
221:C
209:P
205:O
201:P
197:P
114:T
107:C
103:t
75:P
20:)
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