2859:
2911:
462:
1808:
121:
42:
4201:(broadly defined as a figure with three vertices connected by curves that are concave to the exterior of the deltoid, making the interior points a non-convex set). The vertices of the deltoid are at the midpoints of the medians; all points inside the deltoid are on three different area bisectors, while all points outside it are on just one.
2344:
2843:
3646:
lies at the intersection of six angle bisectors. These are the internal angle bisectors at two opposite vertex angles, the external angle bisectors (supplementary angle bisectors) at the other two vertex angles, and the external angle bisectors at the angles formed where the
2499:, one draws a circle whose center is the vertex. The circle meets the angle at two points: one on each leg. Using each of these points as a center, draw two circles of the same size. The intersection of the circles (two points) determines a line that is the angle bisector.
4085:
are the line segments that connect the midpoints of opposite sides, hence each bisecting two sides. The two bimedians and the line segment joining the midpoints of the diagonals are concurrent at a point called the "vertex centroid" and are all bisected by this point.
3574:
4398:
A plane that divides two opposite edges of a tetrahedron in a given ratio also divides the volume of the tetrahedron in the same ratio. Thus any plane containing a bimedian (connector of opposite edges' midpoints) of a tetrahedron bisects the volume of the tetrahedron
1473:
1971:
1197:
4154:; indeed, they are the only area bisectors that go through the centroid. Three other area bisectors are parallel to the triangle's sides; each of these intersects the other two sides so as to divide them into segments with the proportions
3730:
bisector of a side of a triangle is the segment, falling entirely on and inside the triangle, of the line that perpendicularly bisects that side. The three perpendicular bisectors of a triangle's three sides intersect at the
3711:
if it has uniform density; thus any line through a triangle's centroid and one of its vertices bisects the opposite side. The centroid is twice as close to the midpoint of any one side as it is to the opposite vertex.
2099:
444:
1026:
2656:
886:
4279:
2089:
2896:
Three intersection points, two of them between an interior angle bisector and the opposite side, and the third between the other exterior angle bisector and the opposite side extended, are collinear.
4010:
3922:
3835:
1626:
3746:
the two shortest sides' perpendicular bisectors (extended beyond their opposite triangle sides to the circumcenter) are divided by their respective intersecting triangle sides in equal proportions.
3369:
3178:
1772:
1699:
3110:
1289:
1541:
552:
4118:
4089:
The four "maltitudes" of a convex quadrilateral are the perpendiculars to a side through the midpoint of the opposite side, hence bisecting the latter side. If the quadrilateral is
2648:
1299:
2580:
930:
784:
4326:). There are either one, two, or three of these for any given triangle. A line through the incenter bisects one of the area or perimeter if and only if it also bisects the other.
1845:
1071:
1061:
4190:. These six lines are concurrent three at a time: in addition to the three medians being concurrent, any one median is concurrent with two of the side-parallel area bisectors.
4188:
3703:
and the midpoint of the opposite side, so it bisects that side (though not in general perpendicularly). The three medians intersect each other at a point which is called the
3269:
244:
4042:
3334:
3019:
2451:
4212:
to the extended sides of the triangle. The ratio of the area of the envelope of area bisectors to the area of the triangle is invariant for all triangles, and equals
2926:'s side is divided into by a line that bisects the opposite angle. It equates their relative lengths to the relative lengths of the other two sides of the triangle.
3361:
3208:
2964:
554:, whose centers are the endpoints of the segment. The line determined by the points of intersection of the two circles is the perpendicular bisector of the segment.
810:
709:
4065:
2378:
732:
501:
161:
4315:
of a triangle is a line segment having one endpoint at one of the three vertices of the triangle and bisecting the perimeter. The three splitters concur at the
3039:
2398:
683:
663:
643:
620:
594:
574:
181:
263:
3735:(the center of the circle through the three vertices). Thus any line through a triangle's circumcenter and perpendicular to a side bisects that side.
2488:, or line segment that divides an angle of less than 180° into two equal angles. The 'exterior' or 'external bisector' is the line that divides the
4322:
Any line through a triangle that splits both the triangle's area and its perimeter in half goes through the triangle's incenter (the center of its
2339:{\displaystyle \quad (a_{1}-b_{1})x+(a_{2}-b_{2})y+(a_{3}-b_{3})z={\tfrac {1}{2}}(a_{1}^{2}-b_{1}^{2}+a_{2}^{2}-b_{2}^{2}+a_{3}^{2}-b_{3}^{2})\;.}
2462:
2492:(of 180° minus the original angle), formed by one side forming the original angle and the extension of the other side, into two equal angles.
275:
470:
45:
Line DE bisects line AB at D, line EF is a perpendicular bisector of segment AD at C, and line EF is the interior bisector of right angle AED
939:
2838:{\displaystyle {\frac {l_{1}x+m_{1}y+n_{1}}{\sqrt {l_{1}^{2}+m_{1}^{2}}}}=\pm {\frac {l_{2}x+m_{2}y+n_{2}}{\sqrt {l_{2}^{2}+m_{2}^{2}}}}.}
815:
4215:
1978:
3569:{\displaystyle {\frac {(b+c)^{2}}{bc}}t_{a}^{2}+{\frac {(c+a)^{2}}{ca}}t_{b}^{2}+{\frac {(a+b)^{2}}{ab}}t_{c}^{2}=(a+b+c)^{2}.}
3927:
3839:
3752:
1548:
3121:
1704:
1631:
4491:
3047:
4727:
4114:), then the perpendicular to a side from the point of intersection of the diagonals always bisects the opposite side.
4104:
3643:
2481:
measures. An angle only has one bisector. Each point of an angle bisector is equidistant from the sides of the angle.
1204:
3742:
the circumcenter divides the interior perpendicular bisectors of the two shortest sides in equal proportions. In an
2506:(dividing it into three equal parts) cannot be achieved with the compass and ruler alone (this was first proved by
2502:
The proof of the correctness of this construction is fairly intuitive, relying on the symmetry of the problem. The
4383:
If a line segment connecting the diagonals of a quadrilateral bisects both diagonals, then this line segment (the
4460:
Oxman, Victor. "On the existence of triangles with given lengths of one side and two adjacent angle bisectors",
3673:
at any point bisects the angle between the line joining the point to the focus and the line from the point and
3619:
2496:
1468:{\displaystyle \quad (a_{1}-b_{1})x+(a_{2}-b_{2})y={\tfrac {1}{2}}(a_{1}^{2}-b_{1}^{2}+a_{2}^{2}-b_{2}^{2})\;.}
4100:
1484:
506:
1966:{\displaystyle \quad {\vec {x}}\cdot ({\vec {a}}-{\vec {b}})={\tfrac {1}{2}}({\vec {a}}^{2}-{\vec {b}}^{2}).}
1192:{\displaystyle \quad {\vec {x}}\cdot ({\vec {a}}-{\vec {b}})={\tfrac {1}{2}}({\vec {a}}^{2}-{\vec {b}}^{2}).}
2585:
94:
35:
4505:
Oxman, Victor, "A purely geometric proof of the uniqueness of a triangle with prescribed angle bisectors",
2520:
891:
745:
4489:
Mironescu, P., and
Panaitopol, L., "The existence of a triangle with prescribed angle bisector lengths",
4194:
2905:
2478:
266:
1031:
58:
4607:
4157:
4090:
3607:
3282:
are the side lengths opposite vertices B and C; and the side opposite A is divided in the proportion
3224:
196:
27:
4312:
4292:
of the triangle and has one endpoint at the midpoint of one of the three sides. The three cleavers
4015:
2489:
1787:
3296:
2969:
4711:
4588:
4285:
2403:
556:
Because the construction of the bisector is done without the knowledge of the segment's midpoint
62:
2910:
4146:
of the triangle (which connect the sides' midpoints with the opposite vertices), and these are
4687:
4347:
4143:
4121:
forms a quadrilateral from the perpendicular bisectors of the sides of another quadrilateral.
3700:
3696:
4580:
4351:
4293:
4147:
4094:
3615:
3611:
3588:
3579:
No two non-congruent triangles share the same set of three internal angle bisector lengths.
2863:
2503:
98:
31:
3339:
3186:
2937:
4414:
4305:
3743:
1800:
86:
789:
688:
4669:
4047:
2360:
714:
483:
461:
143:
4648:
4539:
4354:
through the center bisect the area and perimeter. In the case of a circle they are the
4346:
All area bisectors and perimeter bisectors of a circle or other ellipse go through the
4297:
4079:
3739:
3708:
3600:
3024:
2879:
2875:
2507:
2485:
2461:
2383:
668:
648:
628:
605:
579:
559:
474:
166:
4504:
4465:
252:
4721:
4415:
Weisstein, Eric W. "Exterior Angle
Bisector." From MathWorld--A Wolfram Web Resource.
4388:
4335:
4198:
4108:
4082:
4075:
3727:
3674:
3648:
3603:
2886:
2514:
1822:
933:
130:
4675:
4652:
3732:
3721:
2890:
1807:
1782:
Perpendicular line segment bisectors were used solving various geometric problems:
78:
4690:
4679:
4520:
4657:
4621:
4384:
4316:
2885:
Three intersection points, each of an external angle bisector with the opposite
184:
4663:
120:
4707:
4209:
20:
4695:
4519:
Weisstein, Eric W. "Quadrilateral." From MathWorld--A Wolfram Web
Resource.
4289:
4205:
4111:
4620:
Weisstein, Eric W. "Tetrahedron." From MathWorld--A Wolfram Web
Resource.
4534:
Mitchell, Douglas W. (2013), "Perpendicular
Bisectors of Triangle Sides",
3210:
and if this bisector divides the side opposite A into segments of lengths
4372:
4355:
4323:
4301:
4151:
4139:
3704:
3670:
3660:
2923:
2867:
1830:
1794:
936:
of the perpendicular line segment bisector. Hence its vector equation is
134:
74:
50:
2650:
then the internal and external bisectors are given by the two equations
439:{\displaystyle |XA|^{2}=|XM|^{2}+|MA|^{2}=|XM|^{2}+|MB|^{2}=|XB|^{2}\;.}
61:
parts (having the same shape and size). Usually it involves a bisecting
4592:
4202:
3666:
3631:
3610:(that is, the four intersection points of adjacent angle bisectors are
711:, and the perpendicular to be constructed is the one bisecting segment
1021:{\displaystyle ({\vec {x}}-{\vec {m}})\cdot ({\vec {a}}-{\vec {b}})=0}
41:
2919:
4584:
2858:
133:
bisector of a line segment is a line which meets the segment at its
3749:
For any triangle the interior perpendicular bisectors are given by
2909:
2857:
2517:. If the angle is formed by the two lines given algebraically as
2474:
2460:
1806:
460:
453:
is usually used for the construction of a perpendicular bisector:
119:
90:
40:
4571:
Dunn, Jas. A.; Pretty, Jas. E. (May 1972). "Halving a triangle".
4135:
1837:
Its vector equation is literally the same as in the plane case:
881:{\displaystyle M:{\vec {m}}={\tfrac {{\vec {a}}+{\vec {b}}}{2}}}
3183:
If the internal bisector of angle A in triangle ABC has length
2484:
The 'interior' or 'internal bisector' of an angle is the line,
3293:
If the internal bisectors of angles A, B, and C have lengths
4274:{\displaystyle {\tfrac {3}{4}}\log _{e}(2)-{\tfrac {1}{2}},}
503:
is bisected by drawing intersecting circles of equal radius
2084:{\displaystyle A=(a_{1},a_{2},a_{3}),B=(b_{1},b_{2},b_{3})}
4706:
This article incorporates material from Angle bisector on
2918:
The angle bisector theorem is concerned with the relative
596:
as the intersection of the bisector and the line segment.
4097:
at (all meet at) a common point called the "anticenter".
3041:, then the length of the internal bisector of angle A is
4005:{\displaystyle p_{c}={\tfrac {2cT}{a^{2}-b^{2}+c^{2}}},}
3917:{\displaystyle p_{b}={\tfrac {2bT}{a^{2}+b^{2}-c^{2}}},}
3830:{\displaystyle p_{a}={\tfrac {2aT}{a^{2}+b^{2}-c^{2}}},}
1621:{\displaystyle \;m=-{\tfrac {b_{1}-a_{1}}{b_{2}-a_{2}}}}
1063:
and expanding the equation leads to the vector equation
1803:
boundaries consist of segments of such lines or planes.
255:
69:. The most often considered types of bisectors are the
16:
Division of something into two equal or congruent parts
4257:
4220:
3945:
3857:
3770:
3173:{\displaystyle {\frac {2bc}{b+c}}\cos {\frac {A}{2}}.}
2465:
Bisection of an angle using a compass and straightedge
2210:
1902:
1767:{\displaystyle \;y_{0}={\tfrac {1}{2}}(a_{2}+b_{2})\;}
1723:
1694:{\displaystyle \;x_{0}={\tfrac {1}{2}}(a_{1}+b_{1})\;}
1650:
1563:
1375:
1128:
841:
599:
This construction is in fact used when constructing a
517:
4218:
4160:
4050:
4018:
3930:
3842:
3755:
3372:
3342:
3299:
3227:
3189:
3124:
3050:
3027:
2972:
2940:
2659:
2588:
2523:
2406:
2386:
2363:
2102:
1981:
1848:
1707:
1634:
1551:
1487:
1302:
1207:
1074:
1034:
942:
894:
818:
792:
748:
717:
691:
671:
651:
631:
608:
582:
562:
509:
486:
278:
199:
169:
146:
4540:
http://forumgeom.fau.edu/FG2013volume13/FG201307.pdf
4308:. The cleavers are parallel to the angle bisectors.
2513:
The internal and external bisectors of an angle are
4466:
http://forumgeom.fau.edu/FG2004volume4/FG200425.pdf
3105:{\displaystyle {\frac {2{\sqrt {bcs(s-a)}}}{b+c}},}
473:, whose possibility depends on the ability to draw
4273:
4182:
4059:
4036:
4004:
3916:
3829:
3699:of a triangle is a line segment going through one
3568:
3355:
3328:
3263:
3202:
3172:
3104:
3033:
3013:
2958:
2837:
2642:
2574:
2445:
2392:
2372:
2338:
2083:
1965:
1766:
1693:
1620:
1535:
1467:
1283:
1191:
1055:
1020:
924:
880:
804:
778:
726:
703:
677:
657:
637:
614:
588:
568:
546:
495:
438:
257:
238:
175:
155:
4632:Altshiller-Court, N. "The tetrahedron." Ch. 4 in
4288:of a triangle is a line segment that bisects the
469:In classical geometry, the bisection is a simple
26:For the bisection theorem in measure theory, see
4712:Creative Commons Attribution/Share-Alike License
4134:There is an infinitude of lines that bisect the
3589:integer triangles with a rational angle bisector
4658:Angle Bisector definition. Math Open Reference
4521:http://mathworld.wolfram.com/Quadrilateral.html
2862:The interior angle bisectors of a triangle are
1284:{\displaystyle A=(a_{1},a_{2}),B=(b_{1},b_{2})}
57:is the division of something into two equal or
4664:Line Bisector definition. Math Open Reference
4605:Kodokostas, Dimitrios, "Triangle Equalizers,"
4093:(inscribed in a circle), these maltitudes are
2357:The perpendicular bisector plane of a segment
163:also has the property that each of its points
4622:http://mathworld.wolfram.com/Tetrahedron.html
1816:Perpendicular line segment bisectors in space
140:The perpendicular bisector of a line segment
8:
4676:Animated instructions for bisecting an angle
3618:. In the latter case the quadrilateral is a
2352:(see above) is literally true in space, too:
93:(that divides it into two equal angles). In
576:, the construction is used for determining
97:, bisection is usually done by a bisecting
4530:
4528:
3661:Parabola § Tangent bisection property
2407:
2332:
2091:one gets the equation in coordinate form:
1763:
1708:
1690:
1635:
1552:
1461:
1291:one gets the equation in coordinate form:
432:
4443:
4441:
4439:
4437:
4435:
4256:
4235:
4219:
4217:
4197:of the infinitude of area bisectors is a
4161:
4159:
4103:states that if a cyclic quadrilateral is
4049:
4017:
3989:
3976:
3963:
3944:
3935:
3929:
3901:
3888:
3875:
3856:
3847:
3841:
3814:
3801:
3788:
3769:
3760:
3754:
3557:
3526:
3521:
3500:
3481:
3472:
3467:
3446:
3427:
3418:
3413:
3392:
3373:
3371:
3347:
3341:
3317:
3304:
3298:
3237:
3232:
3226:
3194:
3188:
3157:
3125:
3123:
3057:
3051:
3049:
3026:
3000:
2971:
2939:
2823:
2818:
2805:
2800:
2788:
2772:
2756:
2749:
2734:
2729:
2716:
2711:
2699:
2683:
2667:
2660:
2658:
2625:
2609:
2593:
2587:
2560:
2544:
2528:
2522:
2438:
2427:
2419:
2408:
2405:
2385:
2362:
2323:
2318:
2305:
2300:
2287:
2282:
2269:
2264:
2251:
2246:
2233:
2228:
2209:
2194:
2181:
2159:
2146:
2124:
2111:
2101:
2072:
2059:
2046:
2021:
2008:
1995:
1980:
1951:
1940:
1939:
1929:
1918:
1917:
1901:
1884:
1883:
1869:
1868:
1851:
1850:
1847:
1754:
1741:
1722:
1713:
1706:
1681:
1668:
1649:
1640:
1633:
1608:
1595:
1583:
1570:
1562:
1550:
1527:
1511:
1486:
1452:
1447:
1434:
1429:
1416:
1411:
1398:
1393:
1374:
1359:
1346:
1324:
1311:
1301:
1272:
1259:
1234:
1221:
1206:
1177:
1166:
1165:
1155:
1144:
1143:
1127:
1110:
1109:
1095:
1094:
1077:
1076:
1073:
1036:
1035:
1033:
998:
997:
983:
982:
962:
961:
947:
946:
941:
911:
910:
896:
895:
893:
860:
859:
845:
844:
840:
826:
825:
817:
791:
765:
764:
750:
749:
747:
716:
690:
670:
650:
630:
607:
581:
561:
539:
528:
516:
508:
485:
465:Construction by straight edge and compass
457:Construction by straight edge and compass
426:
421:
409:
400:
395:
383:
374:
369:
357:
348:
343:
331:
322:
317:
305:
296:
291:
279:
277:
254:
231:
220:
212:
201:
198:
168:
145:
4429:, Dover Publications, 2007 (orig. 1957).
2870:of the triangle, as seen in the diagram.
1536:{\displaystyle \quad y=m(x-x_{0})+y_{0}}
547:{\displaystyle r>{\tfrac {1}{2}}|AB|}
124:Perpendicular bisector of a line segment
4566:
4564:
4562:
4560:
4407:
2893:(fall on the same line as each other).
786:are the position vectors of two points
4375:of a parallelogram bisect each other.
4125:Area bisectors and perimeter bisectors
2934:If the side lengths of a triangle are
2643:{\displaystyle l_{2}x+m_{2}y+n_{2}=0,}
477:of equal radii and different centers:
4204:The sides of the deltoid are arcs of
2575:{\displaystyle l_{1}x+m_{1}y+n_{1}=0}
925:{\displaystyle {\vec {a}}-{\vec {b}}}
779:{\displaystyle {\vec {a}},{\vec {b}}}
471:compass and straightedge construction
7:
4338:bisects the area and the perimeter.
4334:Any line through the midpoint of a
4119:perpendicular bisector construction
3681:Bisectors of the sides of a polygon
645:: drawing a circle whose center is
109:Perpendicular line segment bisector
30:. For the root-finding method, see
4281:i.e. 0.019860... or less than 2%.
3599:The internal angle bisectors of a
1793:Construction of the center of the
1056:{\displaystyle {\vec {m}}=\cdots }
601:line perpendicular to a given line
14:
4451:, Dover Publ., 2007 (orig. 1929).
3021:and A is the angle opposite side
1829:, which meets the segment at its
665:such that it intersects the line
85:, a line that passes through the
73:, a line that passes through the
4682:Using a compass and straightedge
2854:Concurrencies and collinearities
1825:bisector of a line segment is a
1786:Construction of the center of a
4183:{\displaystyle {\sqrt {2}}+1:1}
3264:{\displaystyle t_{a}^{2}+mn=bc}
2914:In this diagram, BD:DC = AB:AC.
2103:
1849:
1488:
1303:
1075:
239:{\displaystyle \quad |XA|=|XB|}
200:
4710:, which is licensed under the
4250:
4244:
3707:of the triangle, which is its
3554:
3535:
3497:
3484:
3443:
3430:
3389:
3376:
3080:
3068:
2997:
2979:
2878:and the bisector of the other
2439:
2428:
2420:
2409:
2329:
2221:
2200:
2174:
2165:
2139:
2130:
2104:
2078:
2039:
2027:
1988:
1957:
1945:
1923:
1913:
1895:
1889:
1874:
1865:
1856:
1760:
1734:
1687:
1661:
1517:
1498:
1458:
1386:
1365:
1339:
1330:
1304:
1278:
1252:
1240:
1214:
1183:
1171:
1149:
1139:
1121:
1115:
1100:
1091:
1082:
1041:
1009:
1003:
988:
979:
973:
967:
952:
943:
916:
901:
865:
850:
831:
770:
755:
540:
529:
422:
410:
396:
384:
370:
358:
344:
332:
318:
306:
292:
280:
232:
221:
213:
202:
1:
4492:American Mathematical Monthly
4037:{\displaystyle a\geq b\geq c}
4670:Perpendicular Line Bisector.
4611:83, April 2010, pp. 141-146.
4387:) is itself bisected by the
4298:center of the Spieker circle
3649:extensions of opposite sides
3329:{\displaystyle t_{a},t_{b},}
3014:{\displaystyle s=(a+b+c)/2,}
187:from segment AB's endpoints:
4449:Advanced Euclidean Geometry
3644:ex-tangential quadrilateral
3638:Ex-tangential quadrilateral
3115:or in trigonometric terms,
2922:of the two segments that a
2446:{\displaystyle \;|XA|=|XB|}
4744:
4634:Modern Pure Solid Geometry
4550:Altshiller-Court, Nathan,
4296:at (all pass through) the
3719:
3658:
2903:
25:
18:
3634:bisects opposite angles.
4573:The Mathematical Gazette
4480:93, March 2009, 115-116.
4142:. Three of them are the
3620:tangential quadrilateral
2497:straightedge and compass
2495:To bisect an angle with
19:Not to be confused with
4672:With interactive applet
4666:With interactive applet
4660:With interactive applet
3716:Perpendicular bisectors
812:, then its midpoint is
249:The proof follows from
95:three-dimensional space
36:Bisect (disambiguation)
4362:Bisectors of diagonals
4275:
4184:
4061:
4038:
4006:
3918:
3831:
3570:
3357:
3330:
3265:
3204:
3174:
3106:
3035:
3015:
2960:
2915:
2906:Angle bisector theorem
2900:Angle bisector theorem
2871:
2866:in a point called the
2839:
2644:
2576:
2504:trisection of an angle
2466:
2447:
2394:
2374:
2340:
2085:
1967:
1812:
1768:
1695:
1622:
1537:
1469:
1285:
1193:
1057:
1022:
926:
882:
806:
780:
728:
705:
679:
659:
639:
616:
590:
570:
548:
497:
466:
440:
259:
240:
177:
157:
125:
46:
34:. For other uses, see
4276:
4185:
4101:Brahmagupta's theorem
4062:
4039:
4007:
3919:
3832:
3571:
3358:
3356:{\displaystyle t_{c}}
3331:
3266:
3205:
3203:{\displaystyle t_{a}}
3175:
3107:
3036:
3016:
2961:
2959:{\displaystyle a,b,c}
2913:
2874:The bisectors of two
2861:
2840:
2645:
2577:
2477:into two angles with
2464:
2448:
2395:
2375:
2341:
2086:
1968:
1810:
1769:
1696:
1623:
1538:
1470:
1286:
1194:
1058:
1023:
927:
883:
807:
781:
729:
706:
680:
660:
640:
617:
591:
571:
549:
498:
464:
441:
260:
241:
178:
158:
123:
44:
4608:Mathematics Magazine
4554:, Dover Publ., 2007.
4478:Mathematical Gazette
4216:
4158:
4048:
4016:
4012:where the sides are
3928:
3840:
3753:
3608:cyclic quadrilateral
3370:
3340:
3297:
3225:
3187:
3122:
3048:
3025:
2970:
2966:, the semiperimeter
2938:
2657:
2586:
2521:
2404:
2384:
2361:
2100:
1979:
1846:
1705:
1632:
1549:
1485:
1300:
1205:
1072:
1032:
940:
892:
816:
790:
746:
715:
689:
669:
649:
629:
606:
580:
560:
507:
484:
276:
253:
197:
167:
144:
28:Ham sandwich theorem
4728:Elementary geometry
4536:Forum Geometricorum
4507:Forum Geometricorum
4462:Forum Geometricorum
4447:Johnson, Roger A.,
3642:The excenter of an
3630:Each diagonal of a
3531:
3477:
3423:
3242:
2828:
2810:
2739:
2721:
2490:supplementary angle
2328:
2310:
2292:
2274:
2256:
2238:
1457:
1439:
1421:
1403:
805:{\displaystyle A,B}
704:{\displaystyle A,B}
267:Pythagoras' theorem
4688:Weisstein, Eric W.
4649:The Angle Bisector
4509:8 (2008): 197–200.
4495:101 (1994): 58–60.
4464:4, 2004, 215–218.
4342:Circle and ellipse
4271:
4266:
4229:
4180:
4150:at the triangle's
4060:{\displaystyle T.}
4057:
4034:
4002:
3997:
3914:
3909:
3827:
3822:
3695:Each of the three
3677:to the directrix.
3566:
3517:
3463:
3409:
3353:
3326:
3261:
3228:
3200:
3170:
3102:
3031:
3011:
2956:
2916:
2872:
2835:
2814:
2796:
2725:
2707:
2640:
2572:
2467:
2443:
2390:
2380:has for any point
2373:{\displaystyle AB}
2370:
2336:
2314:
2296:
2278:
2260:
2242:
2224:
2219:
2081:
1963:
1911:
1813:
1764:
1732:
1691:
1659:
1618:
1616:
1533:
1465:
1443:
1425:
1407:
1389:
1384:
1281:
1189:
1137:
1053:
1018:
922:
878:
876:
802:
776:
727:{\displaystyle AB}
724:
701:
675:
655:
635:
612:
586:
566:
544:
526:
496:{\displaystyle AB}
493:
467:
436:
236:
173:
156:{\displaystyle AB}
153:
126:
101:, also called the
47:
4476:Simons, Stuart.
4427:Analytical Conics
4319:of the triangle.
4265:
4228:
4166:
3996:
3908:
3821:
3583:Integer triangles
3515:
3461:
3407:
3165:
3149:
3097:
3083:
3034:{\displaystyle a}
2830:
2829:
2741:
2740:
2393:{\displaystyle X}
2218:
1948:
1926:
1910:
1892:
1877:
1859:
1731:
1658:
1615:
1383:
1174:
1152:
1136:
1118:
1103:
1085:
1044:
1006:
991:
970:
955:
919:
904:
875:
868:
853:
834:
773:
758:
678:{\displaystyle g}
658:{\displaystyle P}
638:{\displaystyle P}
615:{\displaystyle g}
589:{\displaystyle M}
569:{\displaystyle M}
525:
176:{\displaystyle X}
4735:
4701:
4700:
4680:bisecting a line
4637:
4636:: Chelsea, 1979.
4630:
4624:
4618:
4612:
4603:
4597:
4596:
4579:(396): 105–108.
4568:
4555:
4552:College Geometry
4548:
4542:
4532:
4523:
4517:
4511:
4502:
4496:
4487:
4481:
4474:
4468:
4458:
4452:
4445:
4430:
4423:
4417:
4412:
4394:Volume bisectors
4389:vertex centroid.
4280:
4278:
4277:
4272:
4267:
4258:
4240:
4239:
4230:
4221:
4189:
4187:
4186:
4181:
4167:
4162:
4066:
4064:
4063:
4058:
4044:and the area is
4043:
4041:
4040:
4035:
4011:
4009:
4008:
4003:
3998:
3995:
3994:
3993:
3981:
3980:
3968:
3967:
3957:
3946:
3940:
3939:
3923:
3921:
3920:
3915:
3910:
3907:
3906:
3905:
3893:
3892:
3880:
3879:
3869:
3858:
3852:
3851:
3836:
3834:
3833:
3828:
3823:
3820:
3819:
3818:
3806:
3805:
3793:
3792:
3782:
3771:
3765:
3764:
3614:), or they are
3575:
3573:
3572:
3567:
3562:
3561:
3530:
3525:
3516:
3514:
3506:
3505:
3504:
3482:
3476:
3471:
3462:
3460:
3452:
3451:
3450:
3428:
3422:
3417:
3408:
3406:
3398:
3397:
3396:
3374:
3362:
3360:
3359:
3354:
3352:
3351:
3335:
3333:
3332:
3327:
3322:
3321:
3309:
3308:
3270:
3268:
3267:
3262:
3241:
3236:
3209:
3207:
3206:
3201:
3199:
3198:
3179:
3177:
3176:
3171:
3166:
3158:
3150:
3148:
3137:
3126:
3111:
3109:
3108:
3103:
3098:
3096:
3085:
3084:
3058:
3052:
3040:
3038:
3037:
3032:
3020:
3018:
3017:
3012:
3004:
2965:
2963:
2962:
2957:
2882:are concurrent.
2844:
2842:
2841:
2836:
2831:
2827:
2822:
2809:
2804:
2795:
2794:
2793:
2792:
2777:
2776:
2761:
2760:
2750:
2742:
2738:
2733:
2720:
2715:
2706:
2705:
2704:
2703:
2688:
2687:
2672:
2671:
2661:
2649:
2647:
2646:
2641:
2630:
2629:
2614:
2613:
2598:
2597:
2581:
2579:
2578:
2573:
2565:
2564:
2549:
2548:
2533:
2532:
2452:
2450:
2449:
2444:
2442:
2431:
2423:
2412:
2399:
2397:
2396:
2391:
2379:
2377:
2376:
2371:
2345:
2343:
2342:
2337:
2327:
2322:
2309:
2304:
2291:
2286:
2273:
2268:
2255:
2250:
2237:
2232:
2220:
2211:
2199:
2198:
2186:
2185:
2164:
2163:
2151:
2150:
2129:
2128:
2116:
2115:
2090:
2088:
2087:
2082:
2077:
2076:
2064:
2063:
2051:
2050:
2026:
2025:
2013:
2012:
2000:
1999:
1972:
1970:
1969:
1964:
1956:
1955:
1950:
1949:
1941:
1934:
1933:
1928:
1927:
1919:
1912:
1903:
1894:
1893:
1885:
1879:
1878:
1870:
1861:
1860:
1852:
1833:perpendicularly.
1773:
1771:
1770:
1765:
1759:
1758:
1746:
1745:
1733:
1724:
1718:
1717:
1700:
1698:
1697:
1692:
1686:
1685:
1673:
1672:
1660:
1651:
1645:
1644:
1627:
1625:
1624:
1619:
1617:
1614:
1613:
1612:
1600:
1599:
1589:
1588:
1587:
1575:
1574:
1564:
1542:
1540:
1539:
1534:
1532:
1531:
1516:
1515:
1474:
1472:
1471:
1466:
1456:
1451:
1438:
1433:
1420:
1415:
1402:
1397:
1385:
1376:
1364:
1363:
1351:
1350:
1329:
1328:
1316:
1315:
1290:
1288:
1287:
1282:
1277:
1276:
1264:
1263:
1239:
1238:
1226:
1225:
1198:
1196:
1195:
1190:
1182:
1181:
1176:
1175:
1167:
1160:
1159:
1154:
1153:
1145:
1138:
1129:
1120:
1119:
1111:
1105:
1104:
1096:
1087:
1086:
1078:
1062:
1060:
1059:
1054:
1046:
1045:
1037:
1027:
1025:
1024:
1019:
1008:
1007:
999:
993:
992:
984:
972:
971:
963:
957:
956:
948:
931:
929:
928:
923:
921:
920:
912:
906:
905:
897:
887:
885:
884:
879:
877:
871:
870:
869:
861:
855:
854:
846:
842:
836:
835:
827:
811:
809:
808:
803:
785:
783:
782:
777:
775:
774:
766:
760:
759:
751:
733:
731:
730:
725:
710:
708:
707:
702:
684:
682:
681:
676:
664:
662:
661:
656:
644:
642:
641:
636:
621:
619:
618:
613:
595:
593:
592:
587:
575:
573:
572:
567:
553:
551:
550:
545:
543:
532:
527:
518:
502:
500:
499:
494:
445:
443:
442:
437:
431:
430:
425:
413:
405:
404:
399:
387:
379:
378:
373:
361:
353:
352:
347:
335:
327:
326:
321:
309:
301:
300:
295:
283:
264:
262:
261:
258:{\displaystyle }
256:
245:
243:
242:
237:
235:
224:
216:
205:
182:
180:
179:
174:
162:
160:
159:
154:
137:perpendicularly.
71:segment bisector
65:, also called a
32:Bisection method
4743:
4742:
4738:
4737:
4736:
4734:
4733:
4732:
4718:
4717:
4691:"Line Bisector"
4686:
4685:
4645:
4640:
4631:
4627:
4619:
4615:
4604:
4600:
4585:10.2307/3615256
4570:
4569:
4558:
4549:
4545:
4533:
4526:
4518:
4514:
4503:
4499:
4488:
4484:
4475:
4471:
4459:
4455:
4446:
4433:
4424:
4420:
4413:
4409:
4405:
4396:
4381:
4369:
4364:
4358:of the circle.
4344:
4332:
4306:medial triangle
4300:, which is the
4231:
4214:
4213:
4156:
4155:
4132:
4127:
4072:
4046:
4045:
4014:
4013:
3985:
3972:
3959:
3958:
3947:
3931:
3926:
3925:
3897:
3884:
3871:
3870:
3859:
3843:
3838:
3837:
3810:
3797:
3784:
3783:
3772:
3756:
3751:
3750:
3744:obtuse triangle
3724:
3718:
3693:
3688:
3683:
3663:
3657:
3640:
3628:
3597:
3585:
3553:
3507:
3496:
3483:
3453:
3442:
3429:
3399:
3388:
3375:
3368:
3367:
3343:
3338:
3337:
3313:
3300:
3295:
3294:
3223:
3222:
3190:
3185:
3184:
3138:
3127:
3120:
3119:
3086:
3053:
3046:
3045:
3023:
3022:
2968:
2967:
2936:
2935:
2932:
2908:
2902:
2876:exterior angles
2856:
2851:
2784:
2768:
2752:
2751:
2695:
2679:
2663:
2662:
2655:
2654:
2621:
2605:
2589:
2584:
2583:
2556:
2540:
2524:
2519:
2518:
2459:
2402:
2401:
2382:
2381:
2359:
2358:
2353:
2190:
2177:
2155:
2142:
2120:
2107:
2098:
2097:
2068:
2055:
2042:
2017:
2004:
1991:
1977:
1976:
1938:
1916:
1844:
1843:
1818:
1801:Voronoi diagram
1780:
1750:
1737:
1709:
1703:
1702:
1677:
1664:
1636:
1630:
1629:
1604:
1591:
1590:
1579:
1566:
1565:
1547:
1546:
1544:
1523:
1507:
1483:
1482:
1478:
1355:
1342:
1320:
1307:
1298:
1297:
1268:
1255:
1230:
1217:
1203:
1202:
1164:
1142:
1070:
1069:
1030:
1029:
938:
937:
890:
889:
843:
814:
813:
788:
787:
744:
743:
740:
713:
712:
687:
686:
667:
666:
647:
646:
627:
626:
604:
603:
578:
577:
558:
557:
555:
505:
504:
482:
481:
459:
420:
394:
368:
342:
316:
290:
274:
273:
251:
250:
195:
194:
165:
164:
142:
141:
118:
111:
39:
24:
17:
12:
11:
5:
4741:
4739:
4731:
4730:
4720:
4719:
4703:
4702:
4683:
4673:
4667:
4661:
4655:
4644:
4643:External links
4641:
4639:
4638:
4625:
4613:
4598:
4556:
4543:
4524:
4512:
4497:
4482:
4469:
4453:
4431:
4425:Spain, Barry.
4418:
4406:
4404:
4401:
4395:
4392:
4380:
4377:
4368:
4365:
4363:
4360:
4343:
4340:
4331:
4328:
4270:
4264:
4261:
4255:
4252:
4249:
4246:
4243:
4238:
4234:
4227:
4224:
4179:
4176:
4173:
4170:
4165:
4131:
4128:
4126:
4123:
4107:(that is, has
4071:
4068:
4056:
4053:
4033:
4030:
4027:
4024:
4021:
4001:
3992:
3988:
3984:
3979:
3975:
3971:
3966:
3962:
3956:
3953:
3950:
3943:
3938:
3934:
3913:
3904:
3900:
3896:
3891:
3887:
3883:
3878:
3874:
3868:
3865:
3862:
3855:
3850:
3846:
3826:
3817:
3813:
3809:
3804:
3800:
3796:
3791:
3787:
3781:
3778:
3775:
3768:
3763:
3759:
3740:acute triangle
3720:Main article:
3717:
3714:
3709:center of mass
3692:
3689:
3687:
3684:
3682:
3679:
3659:Main article:
3656:
3653:
3639:
3636:
3627:
3624:
3606:either form a
3596:
3593:
3584:
3581:
3577:
3576:
3565:
3560:
3556:
3552:
3549:
3546:
3543:
3540:
3537:
3534:
3529:
3524:
3520:
3513:
3510:
3503:
3499:
3495:
3492:
3489:
3486:
3480:
3475:
3470:
3466:
3459:
3456:
3449:
3445:
3441:
3438:
3435:
3432:
3426:
3421:
3416:
3412:
3405:
3402:
3395:
3391:
3387:
3384:
3381:
3378:
3350:
3346:
3325:
3320:
3316:
3312:
3307:
3303:
3272:
3271:
3260:
3257:
3254:
3251:
3248:
3245:
3240:
3235:
3231:
3197:
3193:
3181:
3180:
3169:
3164:
3161:
3156:
3153:
3147:
3144:
3141:
3136:
3133:
3130:
3113:
3112:
3101:
3095:
3092:
3089:
3082:
3079:
3076:
3073:
3070:
3067:
3064:
3061:
3056:
3030:
3010:
3007:
3003:
2999:
2996:
2993:
2990:
2987:
2984:
2981:
2978:
2975:
2955:
2952:
2949:
2946:
2943:
2931:
2928:
2904:Main article:
2901:
2898:
2880:interior angle
2855:
2852:
2850:
2847:
2846:
2845:
2834:
2826:
2821:
2817:
2813:
2808:
2803:
2799:
2791:
2787:
2783:
2780:
2775:
2771:
2767:
2764:
2759:
2755:
2748:
2745:
2737:
2732:
2728:
2724:
2719:
2714:
2710:
2702:
2698:
2694:
2691:
2686:
2682:
2678:
2675:
2670:
2666:
2639:
2636:
2633:
2628:
2624:
2620:
2617:
2612:
2608:
2604:
2601:
2596:
2592:
2571:
2568:
2563:
2559:
2555:
2552:
2547:
2543:
2539:
2536:
2531:
2527:
2508:Pierre Wantzel
2471:angle bisector
2458:
2457:Angle bisector
2455:
2441:
2437:
2434:
2430:
2426:
2422:
2418:
2415:
2411:
2400:the property:
2389:
2369:
2366:
2335:
2331:
2326:
2321:
2317:
2313:
2308:
2303:
2299:
2295:
2290:
2285:
2281:
2277:
2272:
2267:
2263:
2259:
2254:
2249:
2245:
2241:
2236:
2231:
2227:
2223:
2217:
2214:
2208:
2205:
2202:
2197:
2193:
2189:
2184:
2180:
2176:
2173:
2170:
2167:
2162:
2158:
2154:
2149:
2145:
2141:
2138:
2135:
2132:
2127:
2123:
2119:
2114:
2110:
2106:
2080:
2075:
2071:
2067:
2062:
2058:
2054:
2049:
2045:
2041:
2038:
2035:
2032:
2029:
2024:
2020:
2016:
2011:
2007:
2003:
1998:
1994:
1990:
1987:
1984:
1962:
1959:
1954:
1947:
1944:
1937:
1932:
1925:
1922:
1915:
1909:
1906:
1900:
1897:
1891:
1888:
1882:
1876:
1873:
1867:
1864:
1858:
1855:
1835:
1834:
1817:
1814:
1811:Bisector plane
1805:
1804:
1798:
1797:of a triangle,
1791:
1788:Thales' circle
1779:
1776:
1762:
1757:
1753:
1749:
1744:
1740:
1736:
1730:
1727:
1721:
1716:
1712:
1689:
1684:
1680:
1676:
1671:
1667:
1663:
1657:
1654:
1648:
1643:
1639:
1611:
1607:
1603:
1598:
1594:
1586:
1582:
1578:
1573:
1569:
1561:
1558:
1555:
1530:
1526:
1522:
1519:
1514:
1510:
1506:
1503:
1500:
1497:
1494:
1491:
1477:Or explicitly:
1464:
1460:
1455:
1450:
1446:
1442:
1437:
1432:
1428:
1424:
1419:
1414:
1410:
1406:
1401:
1396:
1392:
1388:
1382:
1379:
1373:
1370:
1367:
1362:
1358:
1354:
1349:
1345:
1341:
1338:
1335:
1332:
1327:
1323:
1319:
1314:
1310:
1306:
1280:
1275:
1271:
1267:
1262:
1258:
1254:
1251:
1248:
1245:
1242:
1237:
1233:
1229:
1224:
1220:
1216:
1213:
1210:
1188:
1185:
1180:
1173:
1170:
1163:
1158:
1151:
1148:
1141:
1135:
1132:
1126:
1123:
1117:
1114:
1108:
1102:
1099:
1093:
1090:
1084:
1081:
1052:
1049:
1043:
1040:
1017:
1014:
1011:
1005:
1002:
996:
990:
987:
981:
978:
975:
969:
966:
960:
954:
951:
945:
918:
915:
909:
903:
900:
874:
867:
864:
858:
852:
849:
839:
833:
830:
824:
821:
801:
798:
795:
772:
769:
763:
757:
754:
739:
736:
723:
720:
700:
697:
694:
685:in two points
674:
654:
634:
611:
585:
565:
542:
538:
535:
531:
524:
521:
515:
512:
492:
489:
458:
455:
447:
446:
435:
429:
424:
419:
416:
412:
408:
403:
398:
393:
390:
386:
382:
377:
372:
367:
364:
360:
356:
351:
346:
341:
338:
334:
330:
325:
320:
315:
312:
308:
304:
299:
294:
289:
286:
282:
234:
230:
227:
223:
219:
215:
211:
208:
204:
189:
188:
172:
152:
149:
138:
117:
114:
110:
107:
83:angle bisector
15:
13:
10:
9:
6:
4:
3:
2:
4740:
4729:
4726:
4725:
4723:
4716:
4715:
4713:
4709:
4698:
4697:
4692:
4689:
4684:
4681:
4677:
4674:
4671:
4668:
4665:
4662:
4659:
4656:
4654:
4650:
4647:
4646:
4642:
4635:
4629:
4626:
4623:
4617:
4614:
4610:
4609:
4602:
4599:
4594:
4590:
4586:
4582:
4578:
4574:
4567:
4565:
4563:
4561:
4557:
4553:
4547:
4544:
4541:
4537:
4531:
4529:
4525:
4522:
4516:
4513:
4510:
4508:
4501:
4498:
4494:
4493:
4486:
4483:
4479:
4473:
4470:
4467:
4463:
4457:
4454:
4450:
4444:
4442:
4440:
4438:
4436:
4432:
4428:
4422:
4419:
4416:
4411:
4408:
4402:
4400:
4393:
4391:
4390:
4386:
4379:Quadrilateral
4378:
4376:
4374:
4367:Parallelogram
4366:
4361:
4359:
4357:
4353:
4349:
4341:
4339:
4337:
4336:parallelogram
4330:Parallelogram
4329:
4327:
4325:
4320:
4318:
4314:
4309:
4307:
4303:
4299:
4295:
4291:
4287:
4282:
4268:
4262:
4259:
4253:
4247:
4241:
4236:
4232:
4225:
4222:
4211:
4207:
4203:
4200:
4196:
4191:
4177:
4174:
4171:
4168:
4163:
4153:
4149:
4145:
4141:
4137:
4129:
4124:
4122:
4120:
4115:
4113:
4110:
4109:perpendicular
4106:
4105:orthodiagonal
4102:
4098:
4096:
4092:
4087:
4084:
4083:quadrilateral
4081:
4077:
4070:Quadrilateral
4069:
4067:
4054:
4051:
4031:
4028:
4025:
4022:
4019:
3999:
3990:
3986:
3982:
3977:
3973:
3969:
3964:
3960:
3954:
3951:
3948:
3941:
3936:
3932:
3911:
3902:
3898:
3894:
3889:
3885:
3881:
3876:
3872:
3866:
3863:
3860:
3853:
3848:
3844:
3824:
3815:
3811:
3807:
3802:
3798:
3794:
3789:
3785:
3779:
3776:
3773:
3766:
3761:
3757:
3747:
3745:
3741:
3736:
3734:
3729:
3728:perpendicular
3726:The interior
3723:
3715:
3713:
3710:
3706:
3702:
3698:
3690:
3685:
3680:
3678:
3676:
3675:perpendicular
3672:
3668:
3662:
3654:
3652:
3650:
3645:
3637:
3635:
3633:
3625:
3623:
3621:
3617:
3613:
3609:
3605:
3604:quadrilateral
3602:
3595:Quadrilateral
3594:
3592:
3590:
3582:
3580:
3563:
3558:
3550:
3547:
3544:
3541:
3538:
3532:
3527:
3522:
3518:
3511:
3508:
3501:
3493:
3490:
3487:
3478:
3473:
3468:
3464:
3457:
3454:
3447:
3439:
3436:
3433:
3424:
3419:
3414:
3410:
3403:
3400:
3393:
3385:
3382:
3379:
3366:
3365:
3364:
3348:
3344:
3323:
3318:
3314:
3310:
3305:
3301:
3291:
3289:
3285:
3281:
3277:
3258:
3255:
3252:
3249:
3246:
3243:
3238:
3233:
3229:
3221:
3220:
3219:
3217:
3213:
3195:
3191:
3167:
3162:
3159:
3154:
3151:
3145:
3142:
3139:
3134:
3131:
3128:
3118:
3117:
3116:
3099:
3093:
3090:
3087:
3077:
3074:
3071:
3065:
3062:
3059:
3054:
3044:
3043:
3042:
3028:
3008:
3005:
3001:
2994:
2991:
2988:
2985:
2982:
2976:
2973:
2953:
2950:
2947:
2944:
2941:
2929:
2927:
2925:
2921:
2912:
2907:
2899:
2897:
2894:
2892:
2888:
2887:extended side
2883:
2881:
2877:
2869:
2865:
2860:
2853:
2848:
2832:
2824:
2819:
2815:
2811:
2806:
2801:
2797:
2789:
2785:
2781:
2778:
2773:
2769:
2765:
2762:
2757:
2753:
2746:
2743:
2735:
2730:
2726:
2722:
2717:
2712:
2708:
2700:
2696:
2692:
2689:
2684:
2680:
2676:
2673:
2668:
2664:
2653:
2652:
2651:
2637:
2634:
2631:
2626:
2622:
2618:
2615:
2610:
2606:
2602:
2599:
2594:
2590:
2569:
2566:
2561:
2557:
2553:
2550:
2545:
2541:
2537:
2534:
2529:
2525:
2516:
2515:perpendicular
2511:
2509:
2505:
2500:
2498:
2493:
2491:
2487:
2482:
2480:
2476:
2472:
2463:
2456:
2454:
2435:
2432:
2424:
2416:
2413:
2387:
2367:
2364:
2356:
2351:
2346:
2333:
2324:
2319:
2315:
2311:
2306:
2301:
2297:
2293:
2288:
2283:
2279:
2275:
2270:
2265:
2261:
2257:
2252:
2247:
2243:
2239:
2234:
2229:
2225:
2215:
2212:
2206:
2203:
2195:
2191:
2187:
2182:
2178:
2171:
2168:
2160:
2156:
2152:
2147:
2143:
2136:
2133:
2125:
2121:
2117:
2112:
2108:
2096:
2092:
2073:
2069:
2065:
2060:
2056:
2052:
2047:
2043:
2036:
2033:
2030:
2022:
2018:
2014:
2009:
2005:
2001:
1996:
1992:
1985:
1982:
1973:
1960:
1952:
1942:
1935:
1930:
1920:
1907:
1904:
1898:
1886:
1880:
1871:
1862:
1853:
1842:
1838:
1832:
1828:
1824:
1823:perpendicular
1820:
1819:
1815:
1809:
1802:
1799:
1796:
1792:
1789:
1785:
1784:
1783:
1777:
1775:
1755:
1751:
1747:
1742:
1738:
1728:
1725:
1719:
1714:
1710:
1682:
1678:
1674:
1669:
1665:
1655:
1652:
1646:
1641:
1637:
1609:
1605:
1601:
1596:
1592:
1584:
1580:
1576:
1571:
1567:
1559:
1556:
1553:
1528:
1524:
1520:
1512:
1508:
1504:
1501:
1495:
1492:
1489:
1481:
1475:
1462:
1453:
1448:
1444:
1440:
1435:
1430:
1426:
1422:
1417:
1412:
1408:
1404:
1399:
1394:
1390:
1380:
1377:
1371:
1368:
1360:
1356:
1352:
1347:
1343:
1336:
1333:
1325:
1321:
1317:
1312:
1308:
1296:
1292:
1273:
1269:
1265:
1260:
1256:
1249:
1246:
1243:
1235:
1231:
1227:
1222:
1218:
1211:
1208:
1199:
1186:
1178:
1168:
1161:
1156:
1146:
1133:
1130:
1124:
1112:
1106:
1097:
1088:
1079:
1068:
1064:
1050:
1047:
1038:
1028:. Inserting
1015:
1012:
1000:
994:
985:
976:
964:
958:
949:
935:
934:normal vector
913:
907:
898:
872:
862:
856:
847:
837:
828:
822:
819:
799:
796:
793:
767:
761:
752:
737:
735:
721:
718:
698:
695:
692:
672:
652:
632:
625:
609:
602:
597:
583:
563:
536:
533:
522:
519:
513:
510:
490:
487:
478:
476:
472:
463:
456:
454:
452:
433:
427:
417:
414:
406:
401:
391:
388:
380:
375:
365:
362:
354:
349:
339:
336:
328:
323:
313:
310:
302:
297:
287:
284:
272:
271:
270:
268:
247:
228:
225:
217:
209:
206:
193:
186:
170:
150:
147:
139:
136:
132:
131:perpendicular
128:
127:
122:
115:
113:
108:
106:
104:
100:
96:
92:
88:
84:
80:
76:
72:
68:
64:
60:
56:
52:
43:
37:
33:
29:
22:
4705:
4704:
4694:
4653:cut-the-knot
4633:
4628:
4616:
4606:
4601:
4576:
4572:
4551:
4546:
4535:
4515:
4506:
4500:
4490:
4485:
4477:
4472:
4461:
4456:
4448:
4426:
4421:
4410:
4397:
4382:
4370:
4345:
4333:
4321:
4310:
4283:
4192:
4133:
4116:
4099:
4088:
4073:
3748:
3737:
3733:circumcenter
3725:
3722:Circumcircle
3694:
3664:
3641:
3629:
3598:
3587:There exist
3586:
3578:
3292:
3287:
3283:
3279:
3275:
3273:
3215:
3211:
3182:
3114:
2933:
2917:
2895:
2884:
2873:
2512:
2501:
2494:
2483:
2473:divides the
2470:
2468:
2354:
2349:
2347:
2094:
2093:
1974:
1840:
1839:
1836:
1826:
1781:
1778:Applications
1479:
1476:
1294:
1293:
1200:
1066:
1065:
741:
623:
600:
598:
480:The segment
479:
468:
450:
448:
248:
191:
190:
112:
102:
82:
70:
66:
54:
48:
4538:13, 53-59.
4385:Newton Line
4317:Nagel point
3651:intersect.
888:and vector
624:given point
185:equidistant
77:of a given
4708:PlanetMath
4403:References
4350:, and any
4210:asymptotic
4206:hyperbolas
4148:concurrent
4095:concurrent
3616:concurrent
2864:concurrent
116:Definition
81:, and the
21:Dissection
4696:MathWorld
4373:diagonals
4356:diameters
4290:perimeter
4254:−
4242:
4208:that are
4112:diagonals
4076:bimedians
4029:≥
4023:≥
3970:−
3895:−
3808:−
3612:concyclic
3155:
3075:−
2891:collinear
2747:±
2486:half-line
2348:Property
2312:−
2276:−
2240:−
2188:−
2153:−
2118:−
1946:→
1936:−
1924:→
1890:→
1881:−
1875:→
1863:⋅
1857:→
1602:−
1577:−
1560:−
1505:−
1441:−
1405:−
1353:−
1318:−
1172:→
1162:−
1150:→
1116:→
1107:−
1101:→
1089:⋅
1083:→
1051:⋯
1042:→
1004:→
995:−
989:→
977:⋅
968:→
959:−
953:→
917:→
908:−
902:→
866:→
851:→
832:→
771:→
756:→
738:Equations
449:Property
59:congruent
55:bisection
4722:Category
4324:incircle
4313:splitter
4302:incircle
4195:envelope
4152:centroid
4140:triangle
4130:Triangle
4074:The two
3705:centroid
3686:Triangle
3671:parabola
3655:Parabola
2924:triangle
2868:incenter
2849:Triangle
1831:midpoint
1795:Excircle
135:midpoint
103:bisector
75:midpoint
67:bisector
51:geometry
4593:3615256
4304:of the
4286:cleaver
4199:deltoid
4144:medians
3697:medians
3691:Medians
3667:tangent
3632:rhombus
3626:Rhombus
3363:, then
3218:, then
2930:Lengths
2920:lengths
79:segment
4591:
4352:chords
4348:center
4294:concur
4091:cyclic
4080:convex
3738:In an
3701:vertex
3601:convex
3274:where
2889:, are
1701:, and
1545:where
89:of an
4589:JSTOR
4138:of a
4078:of a
3669:to a
2479:equal
2475:angle
1975:With
1827:plane
1201:With
932:is a
622:at a
99:plane
91:angle
4678:and
4371:The
4193:The
4136:area
4117:The
3924:and
3665:The
3336:and
3278:and
3214:and
2582:and
2095:(C3)
1821:The
514:>
475:arcs
265:and
129:The
87:apex
63:line
4651:at
4581:doi
4233:log
3152:cos
2510:).
2469:An
2355:(D)
2350:(D)
1841:(V)
1480:(E)
1295:(C)
1067:(V)
742:If
451:(D)
192:(D)
183:is
49:In
4724::
4693:.
4587:.
4577:56
4575:.
4559:^
4527:^
4434:^
4311:A
4284:A
3622:.
3591:.
3290:.
2453:.
1774:.
1628:,
1543:,
734:.
269::
246:.
105:.
53:,
4714:.
4699:.
4595:.
4583::
4269:,
4263:2
4260:1
4251:)
4248:2
4245:(
4237:e
4226:4
4223:3
4178:1
4175::
4172:1
4169:+
4164:2
4055:.
4052:T
4032:c
4026:b
4020:a
4000:,
3991:2
3987:c
3983:+
3978:2
3974:b
3965:2
3961:a
3955:T
3952:c
3949:2
3942:=
3937:c
3933:p
3912:,
3903:2
3899:c
3890:2
3886:b
3882:+
3877:2
3873:a
3867:T
3864:b
3861:2
3854:=
3849:b
3845:p
3825:,
3816:2
3812:c
3803:2
3799:b
3795:+
3790:2
3786:a
3780:T
3777:a
3774:2
3767:=
3762:a
3758:p
3564:.
3559:2
3555:)
3551:c
3548:+
3545:b
3542:+
3539:a
3536:(
3533:=
3528:2
3523:c
3519:t
3512:b
3509:a
3502:2
3498:)
3494:b
3491:+
3488:a
3485:(
3479:+
3474:2
3469:b
3465:t
3458:a
3455:c
3448:2
3444:)
3440:a
3437:+
3434:c
3431:(
3425:+
3420:2
3415:a
3411:t
3404:c
3401:b
3394:2
3390:)
3386:c
3383:+
3380:b
3377:(
3349:c
3345:t
3324:,
3319:b
3315:t
3311:,
3306:a
3302:t
3288:c
3286::
3284:b
3280:c
3276:b
3259:c
3256:b
3253:=
3250:n
3247:m
3244:+
3239:2
3234:a
3230:t
3216:n
3212:m
3196:a
3192:t
3168:.
3163:2
3160:A
3146:c
3143:+
3140:b
3135:c
3132:b
3129:2
3100:,
3094:c
3091:+
3088:b
3081:)
3078:a
3072:s
3069:(
3066:s
3063:c
3060:b
3055:2
3029:a
3009:,
3006:2
3002:/
2998:)
2995:c
2992:+
2989:b
2986:+
2983:a
2980:(
2977:=
2974:s
2954:c
2951:,
2948:b
2945:,
2942:a
2833:.
2825:2
2820:2
2816:m
2812:+
2807:2
2802:2
2798:l
2790:2
2786:n
2782:+
2779:y
2774:2
2770:m
2766:+
2763:x
2758:2
2754:l
2744:=
2736:2
2731:1
2727:m
2723:+
2718:2
2713:1
2709:l
2701:1
2697:n
2693:+
2690:y
2685:1
2681:m
2677:+
2674:x
2669:1
2665:l
2638:,
2635:0
2632:=
2627:2
2623:n
2619:+
2616:y
2611:2
2607:m
2603:+
2600:x
2595:2
2591:l
2570:0
2567:=
2562:1
2558:n
2554:+
2551:y
2546:1
2542:m
2538:+
2535:x
2530:1
2526:l
2440:|
2436:B
2433:X
2429:|
2425:=
2421:|
2417:A
2414:X
2410:|
2388:X
2368:B
2365:A
2334:.
2330:)
2325:2
2320:3
2316:b
2307:2
2302:3
2298:a
2294:+
2289:2
2284:2
2280:b
2271:2
2266:2
2262:a
2258:+
2253:2
2248:1
2244:b
2235:2
2230:1
2226:a
2222:(
2216:2
2213:1
2207:=
2204:z
2201:)
2196:3
2192:b
2183:3
2179:a
2175:(
2172:+
2169:y
2166:)
2161:2
2157:b
2148:2
2144:a
2140:(
2137:+
2134:x
2131:)
2126:1
2122:b
2113:1
2109:a
2105:(
2079:)
2074:3
2070:b
2066:,
2061:2
2057:b
2053:,
2048:1
2044:b
2040:(
2037:=
2034:B
2031:,
2028:)
2023:3
2019:a
2015:,
2010:2
2006:a
2002:,
1997:1
1993:a
1989:(
1986:=
1983:A
1961:.
1958:)
1953:2
1943:b
1931:2
1921:a
1914:(
1908:2
1905:1
1899:=
1896:)
1887:b
1872:a
1866:(
1854:x
1790:,
1761:)
1756:2
1752:b
1748:+
1743:2
1739:a
1735:(
1729:2
1726:1
1720:=
1715:0
1711:y
1688:)
1683:1
1679:b
1675:+
1670:1
1666:a
1662:(
1656:2
1653:1
1647:=
1642:0
1638:x
1610:2
1606:a
1597:2
1593:b
1585:1
1581:a
1572:1
1568:b
1557:=
1554:m
1529:0
1525:y
1521:+
1518:)
1513:0
1509:x
1502:x
1499:(
1496:m
1493:=
1490:y
1463:.
1459:)
1454:2
1449:2
1445:b
1436:2
1431:2
1427:a
1423:+
1418:2
1413:1
1409:b
1400:2
1395:1
1391:a
1387:(
1381:2
1378:1
1372:=
1369:y
1366:)
1361:2
1357:b
1348:2
1344:a
1340:(
1337:+
1334:x
1331:)
1326:1
1322:b
1313:1
1309:a
1305:(
1279:)
1274:2
1270:b
1266:,
1261:1
1257:b
1253:(
1250:=
1247:B
1244:,
1241:)
1236:2
1232:a
1228:,
1223:1
1219:a
1215:(
1212:=
1209:A
1187:.
1184:)
1179:2
1169:b
1157:2
1147:a
1140:(
1134:2
1131:1
1125:=
1122:)
1113:b
1098:a
1092:(
1080:x
1048:=
1039:m
1016:0
1013:=
1010:)
1001:b
986:a
980:(
974:)
965:m
950:x
944:(
914:b
899:a
873:2
863:b
857:+
848:a
838:=
829:m
823::
820:M
800:B
797:,
794:A
768:b
762:,
753:a
722:B
719:A
699:B
696:,
693:A
673:g
653:P
633:P
610:g
584:M
564:M
541:|
537:B
534:A
530:|
523:2
520:1
511:r
491:B
488:A
434:.
428:2
423:|
418:B
415:X
411:|
407:=
402:2
397:|
392:B
389:M
385:|
381:+
376:2
371:|
366:M
363:X
359:|
355:=
350:2
345:|
340:A
337:M
333:|
329:+
324:2
319:|
314:M
311:X
307:|
303:=
298:2
293:|
288:A
285:X
281:|
233:|
229:B
226:X
222:|
218:=
214:|
210:A
207:X
203:|
171:X
151:B
148:A
38:.
23:.
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