8255:
7846:
3140:
8250:{\displaystyle {\begin{aligned}&{\text{if reference }}\triangle {\text{ is acute:}}\quad \cos A\ :\,\cos B\ :\,\cos C\\&{\begin{array}{rcccc}{\text{if }}\measuredangle A{\text{ is obtuse:}}&\cos A+\sec B\sec C&:&\cos B-\sec B&:&\cos C-\sec C\\{\text{if }}\measuredangle B{\text{ is obtuse:}}&\cos A-\sec A&:&\cos B+\sec C\sec A&:&\cos C-\sec C\\{\text{if }}\measuredangle C{\text{ is obtuse:}}&\cos A-\sec A&:&\cos B-\sec B&:&\cos C+\sec A\sec B\end{array}}\end{aligned}}}
2728:
7456:
7839:
3135:{\displaystyle f(a,b,c)={\begin{cases}1&\quad {\text{if }}a^{2}>b^{2}+bc+c^{2}&\iff {\text{if }}A>2\pi /3\\0&\quad \!\!\displaystyle {{{\text{if }}b^{2}>c^{2}+ca+a^{2}} \atop {{\text{ or }}c^{2}>a^{2}+ab+b^{2}}}&\iff \!\!\displaystyle {{{\text{if }}B>2\pi /3} \atop {{\text{ or }}C>2\pi /3}}\\\csc(A+{\frac {\pi }{3}})&\quad {\text{otherwise }}&\iff A,B,C>2\pi /3\end{cases}}}
40:
7258:
7646:
5104:
4841:
6670:
This is readily seen to be a triangle center function and (provided the triangle is scalene) the corresponding triangle center is the excenter opposite to the largest vertex angle. The other two excenters can be picked out by similar functions. However, as indicated above only one of the excenters of
8783:
Unlike squares and circles, triangles have many centers. The ancient Greeks found four: incenter, centroid, circumcenter, and orthocenter. A fifth center, found much later, is the Fermat point. Thereafter, points now called nine-point center, symmedian point, Gergonne point, and
Feuerbach point, to
3307:
The first and second
Brocard points are one of many bicentric pairs of points, pairs of points defined from a triangle with the property that the pair (but not each individual point) is preserved under similarities of the triangle. Several binary operations, such as midpoint and trilinear product,
5279:
In the following table of more recent triangle centers, no specific notations are mentioned for the various points. Also for each center only the first trilinear coordinate f(a,b,c) is specified. The other coordinates can be easily derived using the cyclicity property of trilinear coordinates.
7451:{\displaystyle f(a,b,c)={\begin{cases}\alpha &\quad {\text{if }}a<b{\text{ and }}a<c&(a{\text{ is least}}),\\\gamma &\quad {\text{if }}a>b{\text{ and }}a>c&(a{\text{ is greatest}}),\\\beta &\quad {\text{otherwise}}&(a{\text{ is in the middle}}).\end{cases}}}
3149:
is bisymmetric and homogeneous so it is a triangle center function. Moreover, the corresponding triangle center coincides with the obtuse angled vertex whenever any vertex angle exceeds 2π/3, and with the 1st isogonic center otherwise. Therefore, this triangle center is none other than the
6666:
6534:
7834:{\displaystyle f(a,b,c)={\begin{cases}\cos A&{\text{if }}\triangle {\text{ is acute}},\\\cos A+\sec B\sec C&{\text{if }}\measuredangle A{\text{ is obtuse}},\\\cos A-\sec A&{\text{if either}}\measuredangle B{\text{ or }}\measuredangle C{\text{ is obtuse}}.\end{cases}}}
4888:
4625:
2706:
8619:
triangle, bisymmetry ensures that all triangle centers are invariant under reflection. Since rotations and translations may be regarded as double reflections they too must preserve triangle centers. These invariance properties provide justification for the definition.
2526:
2336:
751:
This definition ensures that triangle centers of similar triangles meet the invariance criteria specified above. By convention only the first of the three trilinear coordinates of a triangle center is quoted since the other two are obtained by
250:, all triangle centers coincide at its centroid. However the triangle centers generally take different positions from each other on all other triangles. The definitions and properties of thousands of triangle centers have been collected in the
1637:
4582:
1460:
5264:
6551:
7151:
6396:
239:), the center of the transformed triangle is the same point as the transformed center of the original triangle. This invariance is the defining property of a triangle center. It rules out other well-known points such as the
264:
Even though the ancient Greeks discovered the classic centers of a triangle, they had not formulated any definition of a triangle center. After the ancient Greeks, several special points associated with a triangle like the
1962:
6097:
5785:
3948:
4074:
7157:
under addition, subtraction, and multiplication. This gives an easy way to create new triangle centers. However distinct normalized triangle center functions will often define the same triangle center, for example
6781:
3302:
3236:
2341:
2061:
5099:{\displaystyle {\begin{aligned}\sec(A-{\tfrac {\pi }{3}}):\sec(B-{\tfrac {\pi }{3}}):\sec(C-{\tfrac {\pi }{3}})\\\sec(A+{\tfrac {\pi }{3}}):\sec(B+{\tfrac {\pi }{3}}):\sec(C+{\tfrac {\pi }{3}})\end{aligned}}}
4836:{\displaystyle {\begin{aligned}\sin(A+{\tfrac {\pi }{3}}):\sin(B+{\tfrac {\pi }{3}}):\sin(C+{\tfrac {\pi }{3}})\\\sin(A-{\tfrac {\pi }{3}}):\sin(B-{\tfrac {\pi }{3}}):\sin(C-{\tfrac {\pi }{3}})\end{aligned}}}
3238:
These coordinates satisfy the properties of homogeneity and cyclicity but not bisymmetry. So the first
Brocard point is not (in general) a triangle center. The second Brocard point has trilinear coordinates:
2346:
2075:
288:
During the revival of interest in triangle geometry in the 1980s it was noticed that these special points share some general properties that now form the basis for a formal definition of triangle center.
7851:
6903:
2070:
1253:
7626:
it follows that triangle centers could equally well have been defined in terms of barycentrics instead of trilinears. In practice it isn't difficult to switch from one coordinate system to the other.
5635:
2589:
5108:
Intersection of the lines connecting each vertex to the center of an equilateral triangle pointed outwards (first
Napoleon point) or inwards (second Napoleon point), mounted on the opposite side.
8260:
Routine calculation shows that in every case these trilinears represent the incenter of the tangential triangle. So this point is a triangle center that is a close companion of the circumcenter.
7616:
8597:
8483:
6401:
4893:
4630:
4440:
8784:
name a few, were added to the literature. In the 1980s, it was noticed that these special points share some general properties that now form the basis for a formal definition of triangle center
8365:
5560:
3749:
832:
7001:
5929:
1479:
5395:
746:
547:
5846:
7236:
4310:
4186:
1135:
8314:
7492:. Thus every point is potentially a triangle center. However the vast majority of triangle centers are of little interest, just as most continuous functions are of little interest.
1329:
5689:
5476:
3633:
3549:
632:
1045:
6163:
In honor of Clark
Kimberling who created the online encyclopedia of more than 32,000 triangle centers, the triangle centers listed in the encyclopedia are collectively called
1178:
1001:
968:
889:
6303:
6143:
6001:
2532:
is a triangle center function. Since the corresponding triangle center has the same trilinears as the circumcenter, it follows that the circumcenter is a triangle center.
6540:. For an equilateral triangle all three components are equal so all centers coincide with the centroid. So, like a circle, an equilateral triangle has a unique center.
7033:
5329:
4471:
3461:
937:
1806:
3807:
3386:
2714:, one can verify that they indeed satisfy the defining properties of the coordinates of a triangle center. Hence the 1st isogonic center is also a triangle center.
437:
357:
6334:
5135:
410:
330:
377:
6690:
3242:
3176:
3753:
Center of the circle passing through the midpoint of each side, the foot of each altitude, and the midpoint between the orthocenter and each vertex.
891:
both correspond to the centroid. Two triangle center functions define the same triangle center if and only if their ratio is a function symmetric in
6025:
5713:
1005:
are both rational and 1 otherwise. Then for any triangle with integer sides the associated triangle center evaluates to 0:0:0 which is undefined.
898:
Even if a triangle center function is well-defined everywhere the same cannot always be said for its associated triangle center. For example, let
6786:
3843:
6661:{\displaystyle f(a,b,c)={\begin{cases}-1&\quad {\text{if }}a\geq b{\text{ and }}a\geq c,\\\;\;\;1&\quad {\text{otherwise}}.\end{cases}}}
1185:
6536:
so two components of the associated triangle center are always equal. Therefore, all triangle centers of an isosceles triangle must lie on its
3984:
9126:
9080:
205:
99:
7003:
On account of this the definition of triangle center function is sometimes taken to include non-zero homogeneous biantisymmetric functions.
7511:
6529:{\displaystyle {\begin{aligned}f(a,b,c)&=f(b,a,c)&({\text{since }}a=b)\\&=f(b,c,a)&{\text{(by bisymmetry)}}\end{aligned}}}
1969:
1471:
the domain of triangles with an angle exceeding 2π/3 is important; in other words, triangles for which any of the following is true:
2701:{\displaystyle \csc \left(A+{\frac {\pi }{3}}\right):\csc \left(B+{\frac {\pi }{3}}\right):\csc \left(C+{\frac {\pi }{3}}\right).}
8701:
8686:, for instance, can be found for any polygon. Some research has been done on the centers of polygons with more than three sides.
6908:
1057:
295:
252:
5584:
2521:{\displaystyle {\begin{aligned}f(a,c,b)&=a(c^{2}+b^{2}-a^{2})\\&=a(b^{2}+c^{2}-a^{2})\\&=f(a,b,c)\end{aligned}}}
8668:. In non-Euclidean geometry, the assumption that the interior angles of the triangle sum to 180 degrees must be discarded.
7030:. A normalized triangle center function has the same triangle center as the original, and also the stronger property that
7619:
8488:
8374:
4341:
5577:
9289:
8319:
5500:
7916:
3668:
2331:{\displaystyle {\begin{aligned}f(ta,tb,tc)&=ta{\Bigl }\\&=t^{3}{\Bigl }\\&=t^{3}f(a,b,c)\end{aligned}}}
5802:
774:
9305:
8641:
8637:
7682:
7294:
5869:
5353:
656:
9268:
8716:
7634:
There are other center pairs besides the Fermat point and the 1st isogonic center. Another system is formed by
5946:
5652:
5121:
465:
228:
5809:
7165:
4229:
4105:
2764:
1063:
8696:
8287:
3952:
Intersection of the lines connecting each vertex to the point where the incircle touches the opposite side.
232:
212:
4078:
Intersection of the lines connecting each vertex to the point where an excircle touches the opposite side.
8657:
6671:
an isosceles triangle and none of the excenters of an equilateral triangle can ever be a triangle center.
5659:
5493:
5419:
3588:
3554:
3504:
448:
236:
224:
1650:
yet excludes all trivial triangles (i.e. points) and degenerate triangles (i.e. lines) is the set of all
9285:
1269:
There are various instances where it may be desirable to restrict the analysis to a smaller domain than
650:
566:
220:
7478:
is the corresponding triangle center whenever the sides of the reference triangle are labelled so that
3811:
Intersection of the symmedians – the reflection of each median about the corresponding angle bisector.
1018:
2584:
are concurrent and the point of concurrence is the 1st isogonal center. Its trilinear coordinates are
1154:
8917:
7154:
1778:
1146:
1141:
cannot be negative. Furthermore, in order to represent the sides of a triangle they must satisfy the
977:
944:
837:
247:
6587:
6249:
6121:
3308:
when applied to the two
Brocard points, as well as other bicentric pairs, produce triangle centers.
1632:{\displaystyle a^{2}>b^{2}+bc+c^{2};\quad b^{2}>c^{2}+ca+a^{2};\quad c^{2}>a^{2}+ab+b^{2}.}
8661:
6537:
5706:
3638:
1142:
767:
Every triangle center function corresponds to a unique triangle center. This correspondence is not
117:
5953:
9228:
9202:
9171:
9022:
8838:
8816:
8653:
8633:
8629:
8268:
Reflecting a triangle reverses the order of its sides. In the image the coordinates refer to the
4846:
4577:{\displaystyle \csc(A+{\tfrac {\pi }{3}}):\csc(B+{\tfrac {\pi }{3}}):\csc(C+{\tfrac {\pi }{3}}).}
4194:
753:
1261:
is the domain of all triangles, and it is the default domain for all triangle-based functions.
9220:
9163:
9122:
9086:
9076:
8951:
8863:
6114:
5293:
3654:
3466:
3425:
1455:{\displaystyle a^{2}\leq b^{2}+c^{2},\quad b^{2}\leq c^{2}+a^{2},\quad c^{2}\leq a^{2}+b^{2}.}
901:
270:
201:
138:
134:
82:
7641:
and the incenter of the tangential triangle. Consider the triangle center function given by:
5259:{\displaystyle {\frac {bc}{b^{2}-c^{2}}}:{\frac {ca}{c^{2}-a^{2}}}:{\frac {ab}{a^{2}-b^{2}}}}
9278:
9260:
9212:
9155:
9114:
8943:
8834:
8808:
8796:
8768:
8711:
8665:
5862:
4606:
290:
181:
173:
8893:
3780:
3359:
415:
335:
6310:
5346:
4869:
4327:
3766:
3391:
386:
306:
282:
216:
61:
31:
9043:
30:
This article is about a geometry concept. For the place in
Lexington, Kentucky, USA, see
8996:
8772:
6783:
If such a function is also non-zero and homogeneous it is easily seen that the mapping
2573:
be similarly constructed equilateral triangles based on the other two sides of triangle
17:
9249:
8706:
6018:
4210:
3824:
3470:
3168:
1317:
362:
278:
240:
771:. Different functions may define the same triangle center. For example, the functions
9299:
9274:
9232:
9175:
9143:
8970:
3474:
1747:. To support cyclicity it must also be invariant under 2π/3 rotations about the line
274:
243:
which are not invariant under reflection and so fail to qualify as triangle centers.
8931:
4314:
Incenter of the medial triangle. Center of mass of a uniform triangular wireframe.
8947:
8867:
5412:
4457:
3558:
3490:
3151:
266:
189:
103:
9256:
On the geometry of a triangle in the elliptic and in the extended hyperbolic plane
9142:
Al-Sharif, Abdullah; Hajja, Mowaffaq; Krasopoulos, Panagiotis T. (November 2009).
7146:{\displaystyle f(ta,tb,tc)=f(a,b,c)\quad {\text{for all}}\quad t>0,\ (a,b,c).}
1794:
The point of concurrence of the perpendicular bisectors of the sides of triangle
8672:
4091:
3965:
3574:
223:. In other words, for any triangle and any similarity transformation (such as a
197:
121:
39:
9216:
8799:(11 Apr 2018) . "Central Points and Central Lines in the Plane of a Triangle".
8284:
triangle and (using "|" as the separator) the reflection of an arbitrary point
1957:{\displaystyle a(b^{2}+c^{2}-a^{2}):b(c^{2}+a^{2}-b^{2}):c(a^{2}+b^{2}-c^{2}).}
299:
contains an annotated list of over 50,000 triangle centers. Every entry in the
9264:
9191:"Generalization of Kimberling's Concept of Triangle Center for Other Polygons"
9159:
8721:
7153:
Together with the zero function, normalized triangle center functions form an
6092:{\displaystyle \cos {\tfrac {A}{3}}+2\cos {\tfrac {B}{3}}\cos {\tfrac {C}{3}}}
5780:{\displaystyle \sec {\tfrac {A}{2}}\cos {\tfrac {B}{2}}\cos {\tfrac {C}{2}}-1}
149:
9224:
9190:
9167:
9090:
8955:
7843:
For the corresponding triangle center there are four distinct possibilities:
4190:
768:
3943:{\displaystyle {\frac {bc}{b+c-a}}:{\frac {ca}{c+a-b}}:{\frac {ab}{a+b-c}}}
9070:
8683:
8682:
Some centers can be extended to polygons with more than three sides. The
6383:
be a triangle center function. If two sides of a triangle are equal (say
3411:
3395:
3345:
1651:
380:
193:
185:
177:
161:
86:
69:
65:
4586:
Point that is the smallest possible sum of distances from the vertices.
4069:{\displaystyle {\frac {b+c-a}{a}}:{\frac {c+a-b}{b}}:{\frac {a+b-c}{c}}}
8820:
8676:
8371:
is a triangle center function the reflection of its triangle center is
1801:
is the circumcenter. The trilinear coordinates of the circumcenter are
6342:
alone and does not depend on the other angles or on the side lengths.
641:
has both these properties it is called a triangle center function. If
184:
that is in some sense in the middle of the triangle. For example, the
27:
Point in a triangle that can be seen as its middle under some criteria
8741:
actually the 1st isogonic center, but also the Fermat point whenever
6905:
is a triangle center function. The corresponding triangle center is
9072:
Hyperbolic triangle centers : the special relativistic approach
8812:
6776:{\displaystyle f(a,b,c)=-f(a,c,b)\quad {\text{for all}}\quad a,b,c.}
3297:{\displaystyle {\frac {b}{c}}\ :\ {\frac {c}{a}}\ :\ {\frac {a}{b}}}
3231:{\displaystyle {\frac {c}{b}}\ :\ {\frac {a}{c}}\ :\ {\frac {b}{a}}}
9207:
9027:
9021:
Russell, Robert A. (2019-04-18). "Non-Euclidean
Triangle Centers".
8660:. Triangle centers that have the same form for both Euclidean and
2056:{\displaystyle f\left(a,b,c\right)=a\left(b^{2}+c^{2}-a^{2}\right)}
9255:
9189:
Prieto-Martínez, Luis Felipe; Sánchez-Cauce, Raquel (2021-04-02).
6362:
has no trilinear representation using only algebraic functions of
649:
are the side-lengths of a reference triangle then the point whose
38:
9118:
4444:
Point at which the nine-point circle is tangent to the incircle.
9051:
The
Australian Journal of Mathematical Analysis and Applications
9250:
On
Centers and Central Lines of Triangles in the Elliptic Plane
9108:
6246:
if the trilinear coordinates of P can be expressed in the form
8679:
can also be defined, by analogy with 2-dimensional triangles.
8652:
The study of triangle centers traditionally is concerned with
1013:
In some cases these functions are not defined on the whole of
1711:
is a viable domain. In order to support the bisymmetry test
211:
Each of these classical centers has the property that it is
7827:
7444:
6898:{\displaystyle (a,b,c)\to f(a,b,c)^{2}\,f(b,c,a)\,f(c,a,b)}
6654:
3128:
4849:
that transform the triangle into an equilateral triangle.
3173:
The trilinear coordinates of the first Brocard point are:
1248:{\displaystyle a\leq b+c,\quad b\leq c+a,\quad c\leq a+b.}
9110:
Barycentric Calculus in Euclidean and Hyperbolic Geometry
5630:{\displaystyle \tan {\tfrac {A}{2}}+\sec {\tfrac {A}{2}}}
9275:
Triangle Centers in the 2D, 3D, Spherical and Hyperbolic
8920:, Encyclopedia of Triangle Centers, accessed 2012-05-02
379:
is the positional index of the entry. For example, the
8930:
Oakley, Cletus O.; Baker, Justine C. (November 1978).
6078:
6060:
6036:
5827:
5760:
5742:
5724:
5670:
5616:
5595:
5078:
5045:
5012:
4978:
4945:
4912:
4815:
4782:
4749:
4715:
4682:
4649:
4557:
4524:
4491:
1644:
A domain of much practical value since it is dense in
982:
949:
813:
8599:
As this is also the triangle center corresponding to
8491:
8377:
8322:
8290:
7849:
7649:
7514:
7261:
7168:
7036:
6911:
6789:
6693:
6554:
6399:
6313:
6252:
6124:
6028:
5956:
5872:
5812:
5716:
5662:
5587:
5503:
5422:
5356:
5296:
5138:
4891:
4628:
4474:
4344:
4232:
4108:
3987:
3846:
3783:
3671:
3591:
3507:
3428:
3362:
3245:
3179:
2986:
2867:
2731:
2592:
2344:
2073:
1972:
1809:
1482:
1332:
1188:
1157:
1066:
1021:
980:
947:
904:
840:
777:
659:
569:
468:
418:
389:
365:
338:
309:
7611:{\displaystyle a\,f(a,b,c):b\,f(b,c,a):c\,f(c,a,b).}
383:
of a triangle is the second entry and is denoted by
8592:{\displaystyle f(c,b,a)\ |\ f(b,a,c)\ |\ f(a,c,b).}
8478:{\displaystyle f(c,a,b)\ |\ f(b,c,a)\ |\ f(a,b,c),}
5282:
4435:{\displaystyle 1-\cos(B-C):1-\cos(C-A):1-\cos(A-B)}
3320:
8591:
8477:
8359:
8308:
8249:
7833:
7610:
7450:
7230:
7145:
6995:
6897:
6775:
6660:
6528:
6328:
6297:
6137:
6091:
5995:
5923:
5840:
5779:
5683:
5629:
5554:
5470:
5389:
5323:
5258:
5098:
4835:
4576:
4434:
4304:
4180:
4068:
3942:
3801:
3743:
3627:
3543:
3455:
3380:
3296:
3230:
3134:
2700:
2520:
2330:
2055:
1956:
1631:
1464:When differentiating between the Fermat point and
1454:
1247:
1172:
1129:
1039:
995:
962:
931:
883:
826:
740:
626:
541:
431:
404:
371:
351:
324:
9144:"Coincidences of Centers of Plane Quadrilaterals"
2985:
2984:
2866:
2865:
2275:
2223:
2196:
2126:
1654:triangles. It is obtained by removing the planes
8360:{\displaystyle \gamma \ |\ \beta \ |\ \alpha .}
8257:Note that the first is also the circumcenter.
5555:{\displaystyle {\frac {a}{2a^{2}-b^{2}-c^{2}}}}
8839:"This is PART 26: Centers X(50001) – X(52000)"
8656:, but triangle centers can also be studied in
563:Bisymmetry in the second and third variables:
7019:by multiplying it by a symmetric function of
3744:{\displaystyle \cos(B-C):\cos(C-A):\cos(A-B)}
8:
827:{\displaystyle f_{1}(a,b,c)={\tfrac {1}{a}}}
9102:
9100:
6996:{\displaystyle f(a,b,c):f(b,c,a):f(c,a,b).}
5924:{\displaystyle {\frac {a(b+c)^{2}}{b+c-a}}}
7508:and the corresponding triangle center is
6636:
6635:
6634:
5390:{\displaystyle {\frac {1}{\cos B+\cos C}}}
3092:
3088:
2983:
2979:
2829:
2825:
741:{\displaystyle f(a,b,c):f(b,c,a):f(c,a,b)}
9206:
9026:
8554:
8519:
8490:
8440:
8405:
8376:
8343:
8329:
8321:
8289:
8146:
8135:
8038:
8027:
7930:
7919:
7915:
7900:
7884:
7863:
7855:
7850:
7848:
7816:
7805:
7794:
7759:
7748:
7704:
7696:
7677:
7648:
7580:
7549:
7518:
7513:
7504:is a triangle center function then so is
7430:
7417:
7396:
7374:
7360:
7339:
7317:
7303:
7289:
7260:
7216:
7185:
7167:
7098:
7035:
6910:
6870:
6845:
6839:
6788:
6749:
6692:
6643:
6613:
6599:
6582:
6553:
6517:
6464:
6400:
6398:
6312:
6251:
6125:
6123:
6077:
6059:
6035:
6027:
5955:
5895:
5873:
5871:
5826:
5817:
5811:
5759:
5741:
5723:
5715:
5669:
5661:
5615:
5594:
5586:
5543:
5530:
5517:
5504:
5502:
5459:
5446:
5433:
5421:
5357:
5355:
5295:
5247:
5234:
5219:
5207:
5194:
5179:
5167:
5154:
5139:
5137:
5077:
5044:
5011:
4977:
4944:
4911:
4892:
4890:
4814:
4781:
4748:
4714:
4681:
4648:
4629:
4627:
4556:
4523:
4490:
4473:
4343:
4231:
4107:
4042:
4015:
3988:
3986:
3911:
3879:
3847:
3845:
3782:
3670:
3590:
3506:
3427:
3361:
3284:
3265:
3246:
3244:
3218:
3199:
3180:
3178:
3117:
3081:
3065:
3034:
3017:
3016:
3007:
2990:
2989:
2987:
2967:
2945:
2932:
2923:
2922:
2915:
2893:
2880:
2871:
2870:
2868:
2847:
2830:
2817:
2795:
2782:
2773:
2759:
2730:
2710:Expressing these coordinates in terms of
2680:
2645:
2610:
2591:
2471:
2458:
2445:
2416:
2403:
2390:
2345:
2343:
2294:
2274:
2273:
2264:
2251:
2238:
2222:
2221:
2215:
2195:
2194:
2188:
2166:
2144:
2125:
2124:
2074:
2072:
2042:
2029:
2016:
1971:
1942:
1929:
1916:
1894:
1881:
1868:
1846:
1833:
1820:
1808:
1761:. The simplest domain of all is the line
1620:
1598:
1585:
1571:
1549:
1536:
1522:
1500:
1487:
1481:
1443:
1430:
1417:
1403:
1390:
1377:
1363:
1350:
1337:
1331:
1187:
1164:
1160:
1159:
1156:
1117:
1113:
1096:
1092:
1075:
1071:
1065:
1028:
1024:
1023:
1020:
981:
979:
948:
946:
903:
845:
839:
812:
782:
776:
658:
568:
542:{\displaystyle f(ta,tb,tc)=t^{n}f(a,b,c)}
509:
467:
423:
417:
388:
364:
343:
337:
308:
7622:of the triangle center corresponding to
5841:{\displaystyle \sec ^{4}{\tfrac {A}{4}}}
3557:of the sides. Center of the triangle's
2547:be the equilateral triangle having base
8760:
8734:
7231:{\displaystyle (abc)^{-1}(a+b+c)^{3}f.}
4305:{\displaystyle bc(b+c):ca(c+a):ab(a+b)}
4181:{\displaystyle (b+c-a):(c+a-b):(a+b-c)}
1130:{\displaystyle a^{1/2}:b^{1/2}:c^{1/2}}
8309:{\displaystyle \gamma :\beta :\alpha }
4197:(and various equivalent definitions).
9107:Ungar, Abraham Albert (August 2010).
8485:which, by bisymmetry, is the same as
7:
9044:"Hyperbolic Barycentric Coordinates"
5684:{\displaystyle \sec {\tfrac {A}{2}}}
5471:{\displaystyle a(b^{4}+c^{4}-a^{4})}
3628:{\displaystyle \sec A:\sec B:\sec C}
3544:{\displaystyle \cos A:\cos B:\cos C}
1777:which corresponds to the set of all
1145:. So, in practice, every function's
106:(intersect/centered at circumcenter
6375:Isosceles and equilateral triangles
6211:can be expressed as polynomials in
6187:can be expressed as polynomials in
6154:General classes of triangle centers
1717:must be symmetric about the planes
458:may have the following properties:
8648:Non-Euclidean and other geometries
8628:Some other names for dilation are
7860:
7701:
7464:is a triangle center function and
2988:
2869:
645:is a triangle center function and
627:{\displaystyle f(a,b,c)=f(a,c,b).}
25:
8936:The American Mathematical Monthly
7253:be any three real constants. Let
6207:if the trilinear coordinates of
6183:if the trilinear coordinates of
1040:{\displaystyle \mathbb {R} ^{3}.}
43:Five important triangle centers.
9001:MathWorld–A Wolfram Web Resource
8975:MathWorld–A Wolfram Web Resource
8898:MathWorld–A Wolfram Web Resource
8872:MathWorld–A Wolfram Web Resource
8843:Encyclopedia of Triangle Centers
8702:Encyclopedia of Triangle Centers
5268:Various equivalent definitions.
3312:Some well-known triangle centers
1173:{\displaystyle \mathbb {R} ^{3}}
1058:Encyclopedia of Triangle Centers
1056:which is the 365th entry in the
301:Encyclopedia of Triangle Centers
296:Encyclopedia of Triangle Centers
253:Encyclopedia of Triangle Centers
204:, and can be obtained by simple
7868:
7416:
7359:
7302:
7103:
7097:
6754:
6748:
6642:
6598:
3080:
2864:
2772:
1580:
1531:
1412:
1372:
1226:
1207:
1149:is restricted to the region of
1049:For example, the trilinears of
996:{\displaystyle {\tfrac {a}{c}}}
963:{\displaystyle {\tfrac {a}{b}}}
884:{\displaystyle f_{2}(a,b,c)=bc}
8948:10.1080/00029890.1978.11994688
8932:"The Morley Trisector Theorem"
8583:
8565:
8555:
8548:
8530:
8520:
8513:
8495:
8469:
8451:
8441:
8434:
8416:
8406:
8399:
8381:
8344:
8330:
7671:
7653:
7618:Since these are precisely the
7602:
7584:
7571:
7553:
7540:
7522:
7435:
7424:
7401:
7390:
7344:
7333:
7283:
7265:
7213:
7194:
7182:
7169:
7137:
7119:
7094:
7076:
7067:
7040:
6987:
6969:
6960:
6942:
6933:
6915:
6892:
6874:
6867:
6849:
6836:
6817:
6811:
6808:
6790:
6745:
6727:
6715:
6697:
6576:
6558:
6512:
6494:
6478:
6461:
6456:
6438:
6425:
6407:
6356:transcendental triangle center
6346:Transcendental triangle center
6323:
6317:
6298:{\displaystyle f(A):f(B):f(C)}
6292:
6286:
6277:
6271:
6262:
6256:
6138:{\displaystyle {\frac {A}{a}}}
5990:
5963:
5892:
5879:
5465:
5426:
5318:
5300:
5089:
5068:
5056:
5035:
5023:
5002:
4989:
4968:
4956:
4935:
4923:
4902:
4826:
4805:
4793:
4772:
4760:
4739:
4726:
4705:
4693:
4672:
4660:
4639:
4568:
4547:
4535:
4514:
4502:
4481:
4429:
4417:
4399:
4387:
4369:
4357:
4299:
4287:
4275:
4263:
4251:
4239:
4175:
4157:
4151:
4133:
4127:
4109:
3738:
3726:
3714:
3702:
3690:
3678:
3089:
3075:
3056:
2980:
2826:
2753:
2735:
2511:
2493:
2477:
2438:
2422:
2383:
2370:
2352:
2321:
2303:
2270:
2231:
2185:
2175:
2163:
2153:
2141:
2131:
2108:
2081:
1948:
1909:
1900:
1861:
1852:
1813:
926:
908:
869:
851:
806:
788:
735:
717:
708:
690:
681:
663:
618:
600:
591:
573:
536:
518:
499:
472:
399:
393:
319:
313:
1:
7011:Any triangle center function
6230:is the area of the triangle.
748:is called a triangle center.
6118:
6105:
6022:
6009:
5996:{\displaystyle bc(ca+ab-bc)}
5950:
5937:
5866:
5853:
5806:
5793:
5710:
5697:
5656:
5643:
5581:
5578:Congruent isoscelizers point
5568:
5497:
5484:
5416:
5403:
5350:
5337:
3394:. Center of the triangle's
9290:Florida Atlantic University
9286:A Tour of Triangle Geometry
7249:are real variables and let
6338:is a function of the angle
3304:and similar remarks apply.
1316:make specific reference to
760:. This process is known as
9322:
9217:10.1007/s00025-021-01388-4
9069:Ungar, Abraham A. (2010).
9042:Ungar, Abraham A. (2009).
8894:"Triangle Center Function"
6181:polynomial triangle center
6171:Polynomial triangle center
5803:First Ajima-Malfatti point
3317:Classical triangle centers
3166:
221:similarity transformations
48: Reference triangle
29:
9160:10.1007/s00025-009-0417-6
8918:Bicentric Pairs of Points
8264:Bisymmetry and invariance
6675:Biantisymmetric functions
5285:
3323:
9269:University of Evansville
8717:Modern triangle geometry
5947:Equal parallelians point
5653:Yff center of congruence
5324:{\displaystyle f(a,b,c)}
3473:of a uniform triangular
3456:{\displaystyle bc:ca:ab}
2555:on the negative side of
2338:as well as bisymmetric:
1320:, namely that region of
932:{\displaystyle f(a,b,c)}
454:of three real variables
18:Triangle center function
9075:. Dordrecht: Springer.
8997:"Major Triangle Center"
8664:can be expressed using
8624:Alternative terminology
7620:barycentric coordinates
7496:Barycentric coordinates
6195:Regular triangle center
5275:Recent triangle centers
3555:perpendicular bisectors
100:Perpendicular bisectors
68:(intersect/centered at
9195:Results in Mathematics
9148:Results in Mathematics
8675:or higher-dimensional
8658:non-Euclidean geometry
8593:
8479:
8361:
8310:
8251:
7835:
7612:
7452:
7432: is in the middle
7232:
7147:
6997:
6899:
6777:
6662:
6530:
6330:
6299:
6205:regular triangle point
6139:
6093:
5997:
5925:
5842:
5781:
5685:
5631:
5556:
5472:
5391:
5325:
5260:
5100:
4837:
4578:
4436:
4306:
4182:
4070:
3944:
3803:
3745:
3629:
3545:
3457:
3382:
3329:Trilinear coordinates
3298:
3232:
3136:
2702:
2522:
2332:
2057:
1958:
1633:
1456:
1249:
1174:
1131:
1041:
997:
964:
933:
885:
828:
742:
628:
543:
433:
406:
373:
353:
326:
157:
8594:
8480:
8362:
8311:
8252:
7836:
7613:
7453:
7241:Uninteresting centers
7233:
7148:
6998:
6900:
6778:
6663:
6531:
6331:
6300:
6244:major triangle center
6234:Major triangle center
6140:
6115:Hofstadter zero point
6094:
5998:
5926:
5843:
5782:
5686:
5632:
5557:
5473:
5392:
5326:
5261:
5101:
4838:
4579:
4437:
4307:
4183:
4071:
3945:
3804:
3802:{\displaystyle a:b:c}
3746:
3630:
3546:
3458:
3383:
3381:{\displaystyle 1:1:1}
3299:
3233:
3137:
2703:
2523:
2333:
2063:It can be shown that
2058:
1959:
1779:equilateral triangles
1634:
1457:
1250:
1175:
1132:
1042:
998:
965:
934:
886:
829:
743:
651:trilinear coordinates
629:
544:
434:
432:{\displaystyle X_{2}}
407:
374:
354:
352:{\displaystyle X_{n}}
327:
200:were familiar to the
42:
9113:. WORLD SCIENTIFIC.
8801:Mathematics Magazine
8489:
8375:
8320:
8288:
7847:
7647:
7512:
7259:
7166:
7034:
7007:New centers from old
6909:
6787:
6691:
6552:
6397:
6329:{\displaystyle f(X)}
6311:
6250:
6122:
6026:
5954:
5870:
5810:
5714:
5660:
5585:
5501:
5420:
5354:
5294:
5286:ETC reference; Name
5136:
4889:
4626:
4472:
4342:
4230:
4106:
3985:
3844:
3781:
3669:
3637:Intersection of the
3589:
3559:circumscribed circle
3553:Intersection of the
3505:
3465:Intersection of the
3426:
3390:Intersection of the
3360:
3243:
3177:
2729:
2590:
2342:
2071:
1970:
1807:
1480:
1330:
1265:Other useful domains
1186:
1155:
1064:
1019:
978:
945:
902:
838:
775:
657:
567:
466:
449:real-valued function
416:
405:{\displaystyle X(2)}
387:
363:
336:
325:{\displaystyle X(n)}
307:
248:equilateral triangle
8995:Weisstein, Eric W.
8971:"Kimberling Center"
8969:Weisstein, Eric W.
8892:Weisstein, Eric W.
8662:hyperbolic geometry
6350:A triangle center
6199:A triangle center
6175:A triangle center
5707:Isoperimetric point
2536:1st isogonic center
1143:triangle inequality
8773:"Triangle centers"
8654:Euclidean geometry
8589:
8475:
8357:
8306:
8247:
8245:
8241:
7857:if reference
7831:
7826:
7608:
7448:
7443:
7228:
7143:
6993:
6895:
6773:
6658:
6653:
6526:
6524:
6326:
6295:
6238:A triangle center
6165:Kimberling centers
6135:
6089:
6087:
6069:
6045:
5993:
5921:
5838:
5836:
5777:
5769:
5751:
5733:
5681:
5679:
5627:
5625:
5604:
5552:
5468:
5387:
5321:
5256:
5096:
5094:
5087:
5054:
5021:
4987:
4954:
4921:
4833:
4831:
4824:
4791:
4758:
4724:
4691:
4658:
4574:
4566:
4533:
4500:
4432:
4302:
4195:excentral triangle
4178:
4066:
3940:
3799:
3741:
3625:
3541:
3453:
3378:
3294:
3228:
3132:
3127:
3045:
2976:
2698:
2518:
2516:
2328:
2326:
2053:
1954:
1629:
1452:
1245:
1170:
1127:
1037:
993:
991:
960:
958:
929:
881:
824:
822:
754:cyclic permutation
738:
624:
549:for some constant
539:
429:
402:
369:
349:
322:
158:
144:which, along with
9128:978-981-4304-93-1
9082:978-90-481-8637-2
8868:"Triangle Center"
8864:Weisstein, Eric W
8835:Kimberling, Clark
8797:Kimberling, Clark
8769:Kimberling, Clark
8634:isotropic scaling
8561:
8553:
8526:
8518:
8447:
8439:
8412:
8404:
8350:
8342:
8336:
8328:
8149:
8138:
8041:
8030:
7933:
7922:
7896:
7880:
7866:
7858:
7819:
7808:
7797:
7762:
7751:
7707:
7699:
7433:
7420:
7399:
7398: is greatest
7377:
7363:
7342:
7320:
7306:
7118:
7101:
6752:
6646:
6616:
6602:
6520:
6467:
6159:Kimberling center
6151:
6150:
6133:
6086:
6068:
6044:
5919:
5835:
5768:
5750:
5732:
5678:
5624:
5603:
5550:
5385:
5272:
5271:
5254:
5214:
5174:
5086:
5053:
5020:
4986:
4953:
4920:
4823:
4790:
4757:
4723:
4690:
4657:
4607:Isodynamic points
4565:
4532:
4499:
4064:
4037:
4010:
3938:
3906:
3874:
3655:Nine-point center
3292:
3283:
3277:
3273:
3264:
3258:
3254:
3226:
3217:
3211:
3207:
3198:
3192:
3188:
3084:
3073:
3043:
3020:
2993:
2974:
2926:
2874:
2833:
2776:
2688:
2653:
2618:
2580:. Then the lines
1701:Not every subset
990:
957:
821:
443:Formal definition
372:{\displaystyle n}
285:were discovered.
271:nine-point center
139:nine-point center
135:Nine-point circle
16:(Redirected from
9313:
9306:Triangle centers
9279:Wolfram Research
9265:Triangle Centers
9261:Clark Kimberling
9237:
9236:
9210:
9186:
9180:
9179:
9154:(3–4): 231–247.
9139:
9133:
9132:
9104:
9095:
9094:
9066:
9060:
9058:
9048:
9039:
9033:
9032:
9030:
9018:
9012:
9011:
9009:
9007:
8992:
8986:
8985:
8983:
8981:
8966:
8960:
8959:
8927:
8921:
8915:
8909:
8908:
8906:
8904:
8889:
8883:
8882:
8880:
8878:
8860:
8854:
8853:
8851:
8849:
8831:
8825:
8824:
8793:
8787:
8786:
8780:
8779:
8765:
8754:
8739:
8712:Central triangle
8666:gyrotrigonometry
8618:
8603:relative to the
8602:
8598:
8596:
8595:
8590:
8559:
8558:
8551:
8524:
8523:
8516:
8484:
8482:
8481:
8476:
8445:
8444:
8437:
8410:
8409:
8402:
8370:
8366:
8364:
8363:
8358:
8348:
8347:
8340:
8334:
8333:
8326:
8315:
8313:
8312:
8307:
8283:
8256:
8254:
8253:
8248:
8246:
8242:
8150:
8148: is obtuse:
8147:
8139:
8136:
8042:
8040: is obtuse:
8039:
8031:
8028:
7934:
7932: is obtuse:
7931:
7923:
7920:
7913:
7894:
7878:
7867:
7864:
7859:
7856:
7853:
7840:
7838:
7837:
7832:
7830:
7829:
7820:
7817:
7809:
7806:
7798:
7795:
7763:
7760:
7752:
7749:
7708:
7705:
7700:
7697:
7625:
7617:
7615:
7614:
7609:
7507:
7503:
7491:
7477:
7463:
7457:
7455:
7454:
7449:
7447:
7446:
7434:
7431:
7421:
7418:
7400:
7397:
7378:
7375:
7364:
7361:
7343:
7340:
7321:
7318:
7307:
7304:
7252:
7248:
7237:
7235:
7234:
7229:
7221:
7220:
7193:
7192:
7161:
7152:
7150:
7149:
7144:
7116:
7102:
7099:
7029:
7022:
7014:
7002:
7000:
6999:
6994:
6904:
6902:
6901:
6896:
6844:
6843:
6782:
6780:
6779:
6774:
6753:
6750:
6682:
6667:
6665:
6664:
6659:
6657:
6656:
6647:
6644:
6617:
6614:
6603:
6600:
6538:line of symmetry
6535:
6533:
6532:
6527:
6525:
6521:
6518:
6484:
6468:
6465:
6392:
6382:
6365:
6361:
6353:
6341:
6337:
6335:
6333:
6332:
6327:
6304:
6302:
6301:
6296:
6242:is said to be a
6241:
6229:
6225:
6210:
6202:
6190:
6186:
6178:
6144:
6142:
6141:
6136:
6134:
6126:
6098:
6096:
6095:
6090:
6088:
6079:
6070:
6061:
6046:
6037:
6002:
6000:
5999:
5994:
5930:
5928:
5927:
5922:
5920:
5918:
5901:
5900:
5899:
5874:
5863:Apollonius point
5847:
5845:
5844:
5839:
5837:
5828:
5822:
5821:
5786:
5784:
5783:
5778:
5770:
5761:
5752:
5743:
5734:
5725:
5690:
5688:
5687:
5682:
5680:
5671:
5636:
5634:
5633:
5628:
5626:
5617:
5605:
5596:
5561:
5559:
5558:
5553:
5551:
5549:
5548:
5547:
5535:
5534:
5522:
5521:
5505:
5477:
5475:
5474:
5469:
5464:
5463:
5451:
5450:
5438:
5437:
5396:
5394:
5393:
5388:
5386:
5384:
5358:
5330:
5328:
5327:
5322:
5283:
5265:
5263:
5262:
5257:
5255:
5253:
5252:
5251:
5239:
5238:
5228:
5220:
5215:
5213:
5212:
5211:
5199:
5198:
5188:
5180:
5175:
5173:
5172:
5171:
5159:
5158:
5148:
5140:
5130:
5105:
5103:
5102:
5097:
5095:
5088:
5079:
5055:
5046:
5022:
5013:
4988:
4979:
4955:
4946:
4922:
4913:
4883:
4842:
4840:
4839:
4834:
4832:
4825:
4816:
4792:
4783:
4759:
4750:
4725:
4716:
4692:
4683:
4659:
4650:
4620:
4583:
4581:
4580:
4575:
4567:
4558:
4534:
4525:
4501:
4492:
4466:
4441:
4439:
4438:
4433:
4336:
4311:
4309:
4308:
4303:
4224:
4187:
4185:
4184:
4179:
4100:
4075:
4073:
4072:
4067:
4065:
4060:
4043:
4038:
4033:
4016:
4011:
4006:
3989:
3979:
3949:
3947:
3946:
3941:
3939:
3937:
3920:
3912:
3907:
3905:
3888:
3880:
3875:
3873:
3856:
3848:
3838:
3808:
3806:
3805:
3800:
3775:
3750:
3748:
3747:
3742:
3663:
3634:
3632:
3631:
3626:
3583:
3550:
3548:
3547:
3542:
3499:
3462:
3460:
3459:
3454:
3420:
3396:inscribed circle
3387:
3385:
3384:
3379:
3354:
3321:
3303:
3301:
3300:
3295:
3293:
3285:
3281:
3275:
3274:
3266:
3262:
3256:
3255:
3247:
3237:
3235:
3234:
3229:
3227:
3219:
3215:
3209:
3208:
3200:
3196:
3190:
3189:
3181:
3148:
3141:
3139:
3138:
3133:
3131:
3130:
3121:
3085:
3082:
3074:
3066:
3044:
3042:
3038:
3021:
3018:
3015:
3011:
2994:
2991:
2975:
2973:
2972:
2971:
2950:
2949:
2937:
2936:
2927:
2924:
2921:
2920:
2919:
2898:
2897:
2885:
2884:
2875:
2872:
2851:
2834:
2831:
2822:
2821:
2800:
2799:
2787:
2786:
2777:
2774:
2713:
2707:
2705:
2704:
2699:
2694:
2690:
2689:
2681:
2659:
2655:
2654:
2646:
2624:
2620:
2619:
2611:
2583:
2579:
2572:
2565:
2558:
2554:
2550:
2546:
2531:
2527:
2525:
2524:
2519:
2517:
2483:
2476:
2475:
2463:
2462:
2450:
2449:
2428:
2421:
2420:
2408:
2407:
2395:
2394:
2337:
2335:
2334:
2329:
2327:
2299:
2298:
2283:
2279:
2278:
2269:
2268:
2256:
2255:
2243:
2242:
2227:
2226:
2220:
2219:
2204:
2200:
2199:
2193:
2192:
2171:
2170:
2149:
2148:
2130:
2129:
2067:is homogeneous:
2066:
2062:
2060:
2059:
2054:
2052:
2048:
2047:
2046:
2034:
2033:
2021:
2020:
2000:
1996:
1963:
1961:
1960:
1955:
1947:
1946:
1934:
1933:
1921:
1920:
1899:
1898:
1886:
1885:
1873:
1872:
1851:
1850:
1838:
1837:
1825:
1824:
1800:
1776:
1760:
1746:
1736:
1726:
1716:
1710:
1689:
1683:
1673:
1663:
1649:
1638:
1636:
1635:
1630:
1625:
1624:
1603:
1602:
1590:
1589:
1576:
1575:
1554:
1553:
1541:
1540:
1527:
1526:
1505:
1504:
1492:
1491:
1461:
1459:
1458:
1453:
1448:
1447:
1435:
1434:
1422:
1421:
1408:
1407:
1395:
1394:
1382:
1381:
1368:
1367:
1355:
1354:
1342:
1341:
1325:
1274:
1260:
1254:
1252:
1251:
1246:
1181:
1179:
1177:
1176:
1171:
1169:
1168:
1163:
1140:
1136:
1134:
1133:
1128:
1126:
1125:
1121:
1105:
1104:
1100:
1084:
1083:
1079:
1048:
1046:
1044:
1043:
1038:
1033:
1032:
1027:
1004:
1002:
1000:
999:
994:
992:
983:
971:
969:
967:
966:
961:
959:
950:
938:
936:
935:
930:
894:
890:
888:
887:
882:
850:
849:
833:
831:
830:
825:
823:
814:
787:
786:
759:
747:
745:
744:
739:
648:
644:
640:
633:
631:
630:
625:
559:
552:
548:
546:
545:
540:
514:
513:
457:
453:
438:
436:
435:
430:
428:
427:
411:
409:
408:
403:
378:
376:
375:
370:
358:
356:
355:
350:
348:
347:
331:
329:
328:
323:
291:Clark Kimberling
215:(more precisely
154:
147:
143:
132:
126:
115:
109:
97:
91:
80:
74:
59:
54:
47:
21:
9321:
9320:
9316:
9315:
9314:
9312:
9311:
9310:
9296:
9295:
9254:Manfred Evers,
9248:Manfred Evers,
9245:
9240:
9188:
9187:
9183:
9141:
9140:
9136:
9129:
9106:
9105:
9098:
9083:
9068:
9067:
9063:
9046:
9041:
9040:
9036:
9020:
9019:
9015:
9005:
9003:
8994:
8993:
8989:
8979:
8977:
8968:
8967:
8963:
8929:
8928:
8924:
8916:
8912:
8902:
8900:
8891:
8890:
8886:
8876:
8874:
8862:
8861:
8857:
8847:
8845:
8833:
8832:
8828:
8813:10.2307/2690608
8795:
8794:
8790:
8777:
8775:
8767:
8766:
8762:
8758:
8757:
8740:
8736:
8731:
8726:
8692:
8650:
8630:uniform scaling
8626:
8604:
8600:
8487:
8486:
8373:
8372:
8368:
8318:
8317:
8286:
8285:
8269:
8266:
8244:
8243:
8240:
8239:
8207:
8202:
8179:
8174:
8151:
8132:
8131:
8108:
8103:
8071:
8066:
8043:
8024:
8023:
8000:
7995:
7972:
7967:
7935:
7911:
7910:
7865: is acute:
7845:
7844:
7825:
7824:
7818: is obtuse
7792:
7768:
7767:
7761: is obtuse
7746:
7713:
7712:
7694:
7678:
7645:
7644:
7640:
7632:
7623:
7510:
7509:
7505:
7501:
7498:
7479:
7465:
7461:
7442:
7441:
7422:
7414:
7408:
7407:
7388:
7376: and
7357:
7351:
7350:
7331:
7319: and
7300:
7290:
7257:
7256:
7250:
7246:
7243:
7212:
7181:
7164:
7163:
7159:
7032:
7031:
7024:
7020:
7012:
7009:
6907:
6906:
6835:
6785:
6784:
6689:
6688:
6685:biantisymmetric
6680:
6677:
6652:
6651:
6640:
6631:
6630:
6615: and
6596:
6583:
6550:
6549:
6546:
6523:
6522:
6519:(by bisymmetry)
6515:
6482:
6481:
6459:
6428:
6395:
6394:
6384:
6380:
6377:
6372:
6363:
6359:
6351:
6348:
6339:
6309:
6308:
6306:
6248:
6247:
6239:
6236:
6227:
6212:
6208:
6200:
6197:
6188:
6184:
6176:
6173:
6161:
6156:
6120:
6119:
6111:
6024:
6023:
6015:
5952:
5951:
5943:
5902:
5891:
5875:
5868:
5867:
5859:
5813:
5808:
5807:
5799:
5712:
5711:
5703:
5658:
5657:
5649:
5583:
5582:
5574:
5539:
5526:
5513:
5509:
5499:
5498:
5490:
5455:
5442:
5429:
5418:
5417:
5409:
5362:
5352:
5351:
5347:Schiffler point
5343:
5333:Year described
5292:
5291:
5290:
5289:Center function
5277:
5243:
5230:
5229:
5221:
5203:
5190:
5189:
5181:
5163:
5150:
5149:
5141:
5134:
5133:
5126:
5118:
5093:
5092:
4993:
4992:
4887:
4886:
4878:
4874:
4870:Napoleon points
4866:
4860:
4859:
4830:
4829:
4730:
4729:
4624:
4623:
4615:
4611:
4603:
4597:
4596:
4470:
4469:
4462:
4454:
4340:
4339:
4332:
4328:Feuerbach point
4324:
4228:
4227:
4223:
4215:
4207:
4104:
4103:
4096:
4088:
4044:
4017:
3990:
3983:
3982:
3978:
3970:
3962:
3921:
3913:
3889:
3881:
3857:
3849:
3842:
3841:
3837:
3829:
3821:
3779:
3778:
3771:
3767:Symmedian point
3763:
3667:
3666:
3659:
3651:
3587:
3586:
3579:
3571:
3503:
3502:
3495:
3487:
3424:
3423:
3416:
3408:
3392:angle bisectors
3358:
3357:
3350:
3342:
3325:
3324:ETC reference;
3319:
3314:
3241:
3240:
3175:
3174:
3171:
3165:
3160:
3146:
3126:
3125:
3086:
3083:otherwise
3078:
3047:
3046:
2977:
2963:
2941:
2928:
2911:
2889:
2876:
2862:
2856:
2855:
2823:
2813:
2791:
2778:
2770:
2760:
2727:
2726:
2720:
2711:
2673:
2669:
2638:
2634:
2603:
2599:
2588:
2587:
2581:
2574:
2567:
2560:
2556:
2552:
2548:
2541:
2538:
2529:
2515:
2514:
2481:
2480:
2467:
2454:
2441:
2426:
2425:
2412:
2399:
2386:
2373:
2340:
2339:
2325:
2324:
2290:
2281:
2280:
2260:
2247:
2234:
2211:
2202:
2201:
2184:
2162:
2140:
2111:
2069:
2068:
2064:
2038:
2025:
2012:
2011:
2007:
1980:
1976:
1968:
1967:
1938:
1925:
1912:
1890:
1877:
1864:
1842:
1829:
1816:
1805:
1804:
1795:
1792:
1787:
1762:
1748:
1738:
1728:
1718:
1712:
1702:
1699:
1697:Domain symmetry
1685:
1675:
1665:
1655:
1645:
1616:
1594:
1581:
1567:
1545:
1532:
1518:
1496:
1483:
1478:
1477:
1470:
1439:
1426:
1413:
1399:
1386:
1373:
1359:
1346:
1333:
1328:
1327:
1321:
1318:acute triangles
1315:
1308:
1301:
1294:
1287:
1275:. For example:
1270:
1267:
1256:
1184:
1183:
1158:
1153:
1152:
1150:
1138:
1109:
1088:
1067:
1062:
1061:
1055:
1022:
1017:
1016:
1014:
1011:
976:
975:
973:
943:
942:
940:
900:
899:
892:
841:
836:
835:
778:
773:
772:
757:
655:
654:
646:
642:
638:
565:
564:
554:
550:
505:
464:
463:
455:
451:
445:
419:
414:
413:
385:
384:
361:
360:
339:
334:
333:
305:
304:
283:Feuerbach point
262:
170:triangle centre
166:triangle center
156:
152:
145:
141:
130:
128:
124:
113:
111:
107:
95:
93:
89:
78:
76:
72:
62:Angle bisectors
57:
55:
49:
45:
35:
32:Triangle Center
28:
23:
22:
15:
12:
11:
5:
9319:
9317:
9309:
9308:
9298:
9297:
9294:
9293:
9282:
9271:
9258:
9252:
9244:
9243:External links
9241:
9239:
9238:
9181:
9134:
9127:
9096:
9081:
9061:
9034:
9013:
8987:
8961:
8942:(9): 737–745.
8922:
8910:
8884:
8855:
8826:
8807:(3): 163–187.
8788:
8759:
8756:
8755:
8733:
8732:
8730:
8727:
8725:
8724:
8719:
8714:
8709:
8707:Triangle conic
8704:
8699:
8693:
8691:
8688:
8649:
8646:
8625:
8622:
8588:
8585:
8582:
8579:
8576:
8573:
8570:
8567:
8564:
8557:
8550:
8547:
8544:
8541:
8538:
8535:
8532:
8529:
8522:
8515:
8512:
8509:
8506:
8503:
8500:
8497:
8494:
8474:
8471:
8468:
8465:
8462:
8459:
8456:
8453:
8450:
8443:
8436:
8433:
8430:
8427:
8424:
8421:
8418:
8415:
8408:
8401:
8398:
8395:
8392:
8389:
8386:
8383:
8380:
8356:
8353:
8346:
8339:
8332:
8325:
8305:
8302:
8299:
8296:
8293:
8265:
8262:
8238:
8235:
8232:
8229:
8226:
8223:
8220:
8217:
8214:
8211:
8208:
8206:
8203:
8201:
8198:
8195:
8192:
8189:
8186:
8183:
8180:
8178:
8175:
8173:
8170:
8167:
8164:
8161:
8158:
8155:
8152:
8145:
8142:
8134:
8133:
8130:
8127:
8124:
8121:
8118:
8115:
8112:
8109:
8107:
8104:
8102:
8099:
8096:
8093:
8090:
8087:
8084:
8081:
8078:
8075:
8072:
8070:
8067:
8065:
8062:
8059:
8056:
8053:
8050:
8047:
8044:
8037:
8034:
8026:
8025:
8022:
8019:
8016:
8013:
8010:
8007:
8004:
8001:
7999:
7996:
7994:
7991:
7988:
7985:
7982:
7979:
7976:
7973:
7971:
7968:
7966:
7963:
7960:
7957:
7954:
7951:
7948:
7945:
7942:
7939:
7936:
7929:
7926:
7918:
7917:
7914:
7912:
7909:
7906:
7903:
7899:
7893:
7890:
7887:
7883:
7877:
7874:
7871:
7862:
7854:
7852:
7828:
7823:
7815:
7812:
7807: or
7804:
7801:
7793:
7791:
7788:
7785:
7782:
7779:
7776:
7773:
7770:
7769:
7766:
7758:
7755:
7747:
7745:
7742:
7739:
7736:
7733:
7730:
7727:
7724:
7721:
7718:
7715:
7714:
7711:
7706: is acute
7703:
7695:
7693:
7690:
7687:
7684:
7683:
7681:
7676:
7673:
7670:
7667:
7664:
7661:
7658:
7655:
7652:
7638:
7631:
7630:Binary systems
7628:
7607:
7604:
7601:
7598:
7595:
7592:
7589:
7586:
7583:
7579:
7576:
7573:
7570:
7567:
7564:
7561:
7558:
7555:
7552:
7548:
7545:
7542:
7539:
7536:
7533:
7530:
7527:
7524:
7521:
7517:
7497:
7494:
7445:
7440:
7437:
7429:
7426:
7423:
7415:
7413:
7410:
7409:
7406:
7403:
7395:
7392:
7389:
7387:
7384:
7381:
7373:
7370:
7367:
7358:
7356:
7353:
7352:
7349:
7346:
7341: is least
7338:
7335:
7332:
7330:
7327:
7324:
7316:
7313:
7310:
7301:
7299:
7296:
7295:
7293:
7288:
7285:
7282:
7279:
7276:
7273:
7270:
7267:
7264:
7242:
7239:
7227:
7224:
7219:
7215:
7211:
7208:
7205:
7202:
7199:
7196:
7191:
7188:
7184:
7180:
7177:
7174:
7171:
7142:
7139:
7136:
7133:
7130:
7127:
7124:
7121:
7115:
7112:
7109:
7106:
7096:
7093:
7090:
7087:
7084:
7081:
7078:
7075:
7072:
7069:
7066:
7063:
7060:
7057:
7054:
7051:
7048:
7045:
7042:
7039:
7008:
7005:
6992:
6989:
6986:
6983:
6980:
6977:
6974:
6971:
6968:
6965:
6962:
6959:
6956:
6953:
6950:
6947:
6944:
6941:
6938:
6935:
6932:
6929:
6926:
6923:
6920:
6917:
6914:
6894:
6891:
6888:
6885:
6882:
6879:
6876:
6873:
6869:
6866:
6863:
6860:
6857:
6854:
6851:
6848:
6842:
6838:
6834:
6831:
6828:
6825:
6822:
6819:
6816:
6813:
6810:
6807:
6804:
6801:
6798:
6795:
6792:
6772:
6769:
6766:
6763:
6760:
6757:
6747:
6744:
6741:
6738:
6735:
6732:
6729:
6726:
6723:
6720:
6717:
6714:
6711:
6708:
6705:
6702:
6699:
6696:
6676:
6673:
6655:
6650:
6641:
6639:
6633:
6632:
6629:
6626:
6623:
6620:
6612:
6609:
6606:
6597:
6595:
6592:
6589:
6588:
6586:
6581:
6578:
6575:
6572:
6569:
6566:
6563:
6560:
6557:
6545:
6542:
6516:
6514:
6511:
6508:
6505:
6502:
6499:
6496:
6493:
6490:
6487:
6485:
6483:
6480:
6477:
6474:
6471:
6463:
6460:
6458:
6455:
6452:
6449:
6446:
6443:
6440:
6437:
6434:
6431:
6429:
6427:
6424:
6421:
6418:
6415:
6412:
6409:
6406:
6403:
6402:
6376:
6373:
6371:
6368:
6347:
6344:
6325:
6322:
6319:
6316:
6294:
6291:
6288:
6285:
6282:
6279:
6276:
6273:
6270:
6267:
6264:
6261:
6258:
6255:
6235:
6232:
6196:
6193:
6172:
6169:
6160:
6157:
6155:
6152:
6149:
6148:
6145:
6132:
6129:
6117:
6112:
6109:
6103:
6102:
6099:
6085:
6082:
6076:
6073:
6067:
6064:
6058:
6055:
6052:
6049:
6043:
6040:
6034:
6031:
6021:
6016:
6013:
6007:
6006:
6003:
5992:
5989:
5986:
5983:
5980:
5977:
5974:
5971:
5968:
5965:
5962:
5959:
5949:
5944:
5941:
5935:
5934:
5931:
5917:
5914:
5911:
5908:
5905:
5898:
5894:
5890:
5887:
5884:
5881:
5878:
5865:
5860:
5857:
5851:
5850:
5848:
5834:
5831:
5825:
5820:
5816:
5805:
5800:
5797:
5791:
5790:
5787:
5776:
5773:
5767:
5764:
5758:
5755:
5749:
5746:
5740:
5737:
5731:
5728:
5722:
5719:
5709:
5704:
5701:
5695:
5694:
5691:
5677:
5674:
5668:
5665:
5655:
5650:
5647:
5641:
5640:
5637:
5623:
5620:
5614:
5611:
5608:
5602:
5599:
5593:
5590:
5580:
5575:
5572:
5566:
5565:
5562:
5546:
5542:
5538:
5533:
5529:
5525:
5520:
5516:
5512:
5508:
5496:
5491:
5488:
5482:
5481:
5478:
5467:
5462:
5458:
5454:
5449:
5445:
5441:
5436:
5432:
5428:
5425:
5415:
5410:
5407:
5401:
5400:
5397:
5383:
5380:
5377:
5374:
5371:
5368:
5365:
5361:
5349:
5344:
5341:
5335:
5334:
5331:
5320:
5317:
5314:
5311:
5308:
5305:
5302:
5299:
5287:
5276:
5273:
5270:
5269:
5266:
5250:
5246:
5242:
5237:
5233:
5227:
5224:
5218:
5210:
5206:
5202:
5197:
5193:
5187:
5184:
5178:
5170:
5166:
5162:
5157:
5153:
5147:
5144:
5131:
5124:
5119:
5116:
5110:
5109:
5106:
5091:
5085:
5082:
5076:
5073:
5070:
5067:
5064:
5061:
5058:
5052:
5049:
5043:
5040:
5037:
5034:
5031:
5028:
5025:
5019:
5016:
5010:
5007:
5004:
5001:
4998:
4995:
4994:
4991:
4985:
4982:
4976:
4973:
4970:
4967:
4964:
4961:
4958:
4952:
4949:
4943:
4940:
4937:
4934:
4931:
4928:
4925:
4919:
4916:
4910:
4907:
4904:
4901:
4898:
4895:
4894:
4884:
4872:
4867:
4864:
4857:
4851:
4850:
4843:
4828:
4822:
4819:
4813:
4810:
4807:
4804:
4801:
4798:
4795:
4789:
4786:
4780:
4777:
4774:
4771:
4768:
4765:
4762:
4756:
4753:
4747:
4744:
4741:
4738:
4735:
4732:
4731:
4728:
4722:
4719:
4713:
4710:
4707:
4704:
4701:
4698:
4695:
4689:
4686:
4680:
4677:
4674:
4671:
4668:
4665:
4662:
4656:
4653:
4647:
4644:
4641:
4638:
4635:
4632:
4631:
4621:
4609:
4604:
4601:
4594:
4588:
4587:
4584:
4573:
4570:
4564:
4561:
4555:
4552:
4549:
4546:
4543:
4540:
4537:
4531:
4528:
4522:
4519:
4516:
4513:
4510:
4507:
4504:
4498:
4495:
4489:
4486:
4483:
4480:
4477:
4467:
4460:
4455:
4452:
4446:
4445:
4442:
4431:
4428:
4425:
4422:
4419:
4416:
4413:
4410:
4407:
4404:
4401:
4398:
4395:
4392:
4389:
4386:
4383:
4380:
4377:
4374:
4371:
4368:
4365:
4362:
4359:
4356:
4353:
4350:
4347:
4337:
4330:
4325:
4322:
4316:
4315:
4312:
4301:
4298:
4295:
4292:
4289:
4286:
4283:
4280:
4277:
4274:
4271:
4268:
4265:
4262:
4259:
4256:
4253:
4250:
4247:
4244:
4241:
4238:
4235:
4225:
4219:
4213:
4211:Spieker center
4208:
4205:
4199:
4198:
4188:
4177:
4174:
4171:
4168:
4165:
4162:
4159:
4156:
4153:
4150:
4147:
4144:
4141:
4138:
4135:
4132:
4129:
4126:
4123:
4120:
4117:
4114:
4111:
4101:
4094:
4089:
4086:
4080:
4079:
4076:
4063:
4059:
4056:
4053:
4050:
4047:
4041:
4036:
4032:
4029:
4026:
4023:
4020:
4014:
4009:
4005:
4002:
3999:
3996:
3993:
3980:
3974:
3968:
3963:
3960:
3954:
3953:
3950:
3936:
3933:
3930:
3927:
3924:
3919:
3916:
3910:
3904:
3901:
3898:
3895:
3892:
3887:
3884:
3878:
3872:
3869:
3866:
3863:
3860:
3855:
3852:
3839:
3833:
3827:
3825:Gergonne point
3822:
3819:
3813:
3812:
3809:
3798:
3795:
3792:
3789:
3786:
3776:
3769:
3764:
3761:
3755:
3754:
3751:
3740:
3737:
3734:
3731:
3728:
3725:
3722:
3719:
3716:
3713:
3710:
3707:
3704:
3701:
3698:
3695:
3692:
3689:
3686:
3683:
3680:
3677:
3674:
3664:
3657:
3652:
3649:
3643:
3642:
3635:
3624:
3621:
3618:
3615:
3612:
3609:
3606:
3603:
3600:
3597:
3594:
3584:
3577:
3572:
3569:
3563:
3562:
3551:
3540:
3537:
3534:
3531:
3528:
3525:
3522:
3519:
3516:
3513:
3510:
3500:
3493:
3488:
3485:
3479:
3478:
3471:Center of mass
3463:
3452:
3449:
3446:
3443:
3440:
3437:
3434:
3431:
3421:
3414:
3409:
3406:
3400:
3399:
3388:
3377:
3374:
3371:
3368:
3365:
3355:
3348:
3343:
3340:
3334:
3333:
3330:
3327:
3318:
3315:
3313:
3310:
3291:
3288:
3280:
3272:
3269:
3261:
3253:
3250:
3225:
3222:
3214:
3206:
3203:
3195:
3187:
3184:
3169:Brocard points
3167:Main article:
3164:
3163:Brocard points
3161:
3159:
3156:
3143:
3142:
3129:
3124:
3120:
3116:
3113:
3110:
3107:
3104:
3101:
3098:
3095:
3091:
3087:
3079:
3077:
3072:
3069:
3064:
3061:
3058:
3055:
3052:
3049:
3048:
3041:
3037:
3033:
3030:
3027:
3024:
3019: or
3014:
3010:
3006:
3003:
3000:
2997:
2982:
2978:
2970:
2966:
2962:
2959:
2956:
2953:
2948:
2944:
2940:
2935:
2931:
2925: or
2918:
2914:
2910:
2907:
2904:
2901:
2896:
2892:
2888:
2883:
2879:
2863:
2861:
2858:
2857:
2854:
2850:
2846:
2843:
2840:
2837:
2828:
2824:
2820:
2816:
2812:
2809:
2806:
2803:
2798:
2794:
2790:
2785:
2781:
2771:
2769:
2766:
2765:
2763:
2758:
2755:
2752:
2749:
2746:
2743:
2740:
2737:
2734:
2719:
2716:
2697:
2693:
2687:
2684:
2679:
2676:
2672:
2668:
2665:
2662:
2658:
2652:
2649:
2644:
2641:
2637:
2633:
2630:
2627:
2623:
2617:
2614:
2609:
2606:
2602:
2598:
2595:
2537:
2534:
2513:
2510:
2507:
2504:
2501:
2498:
2495:
2492:
2489:
2486:
2484:
2482:
2479:
2474:
2470:
2466:
2461:
2457:
2453:
2448:
2444:
2440:
2437:
2434:
2431:
2429:
2427:
2424:
2419:
2415:
2411:
2406:
2402:
2398:
2393:
2389:
2385:
2382:
2379:
2376:
2374:
2372:
2369:
2366:
2363:
2360:
2357:
2354:
2351:
2348:
2347:
2323:
2320:
2317:
2314:
2311:
2308:
2305:
2302:
2297:
2293:
2289:
2286:
2284:
2282:
2277:
2272:
2267:
2263:
2259:
2254:
2250:
2246:
2241:
2237:
2233:
2230:
2225:
2218:
2214:
2210:
2207:
2205:
2203:
2198:
2191:
2187:
2183:
2180:
2177:
2174:
2169:
2165:
2161:
2158:
2155:
2152:
2147:
2143:
2139:
2136:
2133:
2128:
2123:
2120:
2117:
2114:
2112:
2110:
2107:
2104:
2101:
2098:
2095:
2092:
2089:
2086:
2083:
2080:
2077:
2076:
2051:
2045:
2041:
2037:
2032:
2028:
2024:
2019:
2015:
2010:
2006:
2003:
1999:
1995:
1992:
1989:
1986:
1983:
1979:
1975:
1953:
1950:
1945:
1941:
1937:
1932:
1928:
1924:
1919:
1915:
1911:
1908:
1905:
1902:
1897:
1893:
1889:
1884:
1880:
1876:
1871:
1867:
1863:
1860:
1857:
1854:
1849:
1845:
1841:
1836:
1832:
1828:
1823:
1819:
1815:
1812:
1791:
1788:
1786:
1783:
1698:
1695:
1694:
1693:
1692:
1691:
1628:
1623:
1619:
1615:
1612:
1609:
1606:
1601:
1597:
1593:
1588:
1584:
1579:
1574:
1570:
1566:
1563:
1560:
1557:
1552:
1548:
1544:
1539:
1535:
1530:
1525:
1521:
1517:
1514:
1511:
1508:
1503:
1499:
1495:
1490:
1486:
1475:
1474:
1473:
1472:
1468:
1462:
1451:
1446:
1442:
1438:
1433:
1429:
1425:
1420:
1416:
1411:
1406:
1402:
1398:
1393:
1389:
1385:
1380:
1376:
1371:
1366:
1362:
1358:
1353:
1349:
1345:
1340:
1336:
1313:
1306:
1299:
1292:
1285:
1266:
1263:
1244:
1241:
1238:
1235:
1232:
1229:
1225:
1222:
1219:
1216:
1213:
1210:
1206:
1203:
1200:
1197:
1194:
1191:
1167:
1162:
1124:
1120:
1116:
1112:
1108:
1103:
1099:
1095:
1091:
1087:
1082:
1078:
1074:
1070:
1053:
1036:
1031:
1026:
1010:
1009:Default domain
1007:
989:
986:
956:
953:
928:
925:
922:
919:
916:
913:
910:
907:
880:
877:
874:
871:
868:
865:
862:
859:
856:
853:
848:
844:
820:
817:
811:
808:
805:
802:
799:
796:
793:
790:
785:
781:
737:
734:
731:
728:
725:
722:
719:
716:
713:
710:
707:
704:
701:
698:
695:
692:
689:
686:
683:
680:
677:
674:
671:
668:
665:
662:
637:If a non-zero
635:
634:
623:
620:
617:
614:
611:
608:
605:
602:
599:
596:
593:
590:
587:
584:
581:
578:
575:
572:
561:
538:
535:
532:
529:
526:
523:
520:
517:
512:
508:
504:
501:
498:
495:
492:
489:
486:
483:
480:
477:
474:
471:
444:
441:
426:
422:
401:
398:
395:
392:
368:
346:
342:
321:
318:
315:
312:
303:is denoted by
279:Gergonne point
261:
258:
241:Brocard points
202:ancient Greeks
148:, lies on the
129:
120:(intersect at
112:
94:
85:(intersect at
77:
56:
44:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
9318:
9307:
9304:
9303:
9301:
9291:
9287:
9283:
9280:
9276:
9272:
9270:
9266:
9262:
9259:
9257:
9253:
9251:
9247:
9246:
9242:
9234:
9230:
9226:
9222:
9218:
9214:
9209:
9204:
9200:
9196:
9192:
9185:
9182:
9177:
9173:
9169:
9165:
9161:
9157:
9153:
9149:
9145:
9138:
9135:
9130:
9124:
9120:
9116:
9112:
9111:
9103:
9101:
9097:
9092:
9088:
9084:
9078:
9074:
9073:
9065:
9062:
9059:, article #18
9056:
9052:
9045:
9038:
9035:
9029:
9024:
9017:
9014:
9002:
8998:
8991:
8988:
8976:
8972:
8965:
8962:
8957:
8953:
8949:
8945:
8941:
8937:
8933:
8926:
8923:
8919:
8914:
8911:
8899:
8895:
8888:
8885:
8873:
8869:
8865:
8859:
8856:
8844:
8840:
8836:
8830:
8827:
8822:
8818:
8814:
8810:
8806:
8802:
8798:
8792:
8789:
8785:
8774:
8770:
8764:
8761:
8752:
8748:
8744:
8738:
8735:
8728:
8723:
8720:
8718:
8715:
8713:
8710:
8708:
8705:
8703:
8700:
8698:
8695:
8694:
8689:
8687:
8685:
8680:
8678:
8674:
8669:
8667:
8663:
8659:
8655:
8647:
8645:
8643:
8639:
8635:
8631:
8623:
8621:
8616:
8612:
8608:
8586:
8580:
8577:
8574:
8571:
8568:
8562:
8545:
8542:
8539:
8536:
8533:
8527:
8510:
8507:
8504:
8501:
8498:
8492:
8472:
8466:
8463:
8460:
8457:
8454:
8448:
8431:
8428:
8425:
8422:
8419:
8413:
8396:
8393:
8390:
8387:
8384:
8378:
8354:
8351:
8337:
8323:
8303:
8300:
8297:
8294:
8291:
8281:
8277:
8273:
8263:
8261:
8258:
8236:
8233:
8230:
8227:
8224:
8221:
8218:
8215:
8212:
8209:
8204:
8199:
8196:
8193:
8190:
8187:
8184:
8181:
8176:
8171:
8168:
8165:
8162:
8159:
8156:
8153:
8143:
8140:
8128:
8125:
8122:
8119:
8116:
8113:
8110:
8105:
8100:
8097:
8094:
8091:
8088:
8085:
8082:
8079:
8076:
8073:
8068:
8063:
8060:
8057:
8054:
8051:
8048:
8045:
8035:
8032:
8020:
8017:
8014:
8011:
8008:
8005:
8002:
7997:
7992:
7989:
7986:
7983:
7980:
7977:
7974:
7969:
7964:
7961:
7958:
7955:
7952:
7949:
7946:
7943:
7940:
7937:
7927:
7924:
7907:
7904:
7901:
7897:
7891:
7888:
7885:
7881:
7875:
7872:
7869:
7841:
7821:
7813:
7810:
7802:
7799:
7789:
7786:
7783:
7780:
7777:
7774:
7771:
7764:
7756:
7753:
7743:
7740:
7737:
7734:
7731:
7728:
7725:
7722:
7719:
7716:
7709:
7691:
7688:
7685:
7679:
7674:
7668:
7665:
7662:
7659:
7656:
7650:
7642:
7637:
7629:
7627:
7621:
7605:
7599:
7596:
7593:
7590:
7587:
7581:
7577:
7574:
7568:
7565:
7562:
7559:
7556:
7550:
7546:
7543:
7537:
7534:
7531:
7528:
7525:
7519:
7515:
7495:
7493:
7490:
7486:
7482:
7476:
7472:
7468:
7458:
7438:
7427:
7411:
7404:
7393:
7385:
7382:
7379:
7371:
7368:
7365:
7354:
7347:
7336:
7328:
7325:
7322:
7314:
7311:
7308:
7297:
7291:
7286:
7280:
7277:
7274:
7271:
7268:
7262:
7254:
7240:
7238:
7225:
7222:
7217:
7209:
7206:
7203:
7200:
7197:
7189:
7186:
7178:
7175:
7172:
7156:
7140:
7134:
7131:
7128:
7125:
7122:
7113:
7110:
7107:
7104:
7091:
7088:
7085:
7082:
7079:
7073:
7070:
7064:
7061:
7058:
7055:
7052:
7049:
7046:
7043:
7037:
7027:
7018:
7006:
7004:
6990:
6984:
6981:
6978:
6975:
6972:
6966:
6963:
6957:
6954:
6951:
6948:
6945:
6939:
6936:
6930:
6927:
6924:
6921:
6918:
6912:
6889:
6886:
6883:
6880:
6877:
6871:
6864:
6861:
6858:
6855:
6852:
6846:
6840:
6832:
6829:
6826:
6823:
6820:
6814:
6805:
6802:
6799:
6796:
6793:
6770:
6767:
6764:
6761:
6758:
6755:
6742:
6739:
6736:
6733:
6730:
6724:
6721:
6718:
6712:
6709:
6706:
6703:
6700:
6694:
6686:
6674:
6672:
6668:
6648:
6637:
6627:
6624:
6621:
6618:
6610:
6607:
6604:
6593:
6590:
6584:
6579:
6573:
6570:
6567:
6564:
6561:
6555:
6543:
6541:
6539:
6509:
6506:
6503:
6500:
6497:
6491:
6488:
6486:
6475:
6472:
6469:
6453:
6450:
6447:
6444:
6441:
6435:
6432:
6430:
6422:
6419:
6416:
6413:
6410:
6404:
6391:
6387:
6374:
6370:Miscellaneous
6369:
6367:
6357:
6345:
6343:
6320:
6314:
6289:
6283:
6280:
6274:
6268:
6265:
6259:
6253:
6245:
6233:
6231:
6224:
6220:
6216:
6206:
6194:
6192:
6182:
6170:
6168:
6166:
6158:
6153:
6146:
6130:
6127:
6116:
6113:
6108:
6104:
6100:
6083:
6080:
6074:
6071:
6065:
6062:
6056:
6053:
6050:
6047:
6041:
6038:
6032:
6029:
6020:
6019:Morley center
6017:
6012:
6008:
6004:
5987:
5984:
5981:
5978:
5975:
5972:
5969:
5966:
5960:
5957:
5948:
5945:
5940:
5936:
5932:
5915:
5912:
5909:
5906:
5903:
5896:
5888:
5885:
5882:
5876:
5864:
5861:
5856:
5852:
5849:
5832:
5829:
5823:
5818:
5814:
5804:
5801:
5796:
5792:
5788:
5774:
5771:
5765:
5762:
5756:
5753:
5747:
5744:
5738:
5735:
5729:
5726:
5720:
5717:
5708:
5705:
5700:
5696:
5692:
5675:
5672:
5666:
5663:
5654:
5651:
5646:
5642:
5638:
5621:
5618:
5612:
5609:
5606:
5600:
5597:
5591:
5588:
5579:
5576:
5571:
5567:
5563:
5544:
5540:
5536:
5531:
5527:
5523:
5518:
5514:
5510:
5506:
5495:
5492:
5487:
5483:
5479:
5460:
5456:
5452:
5447:
5443:
5439:
5434:
5430:
5423:
5414:
5411:
5406:
5402:
5398:
5381:
5378:
5375:
5372:
5369:
5366:
5363:
5359:
5348:
5345:
5340:
5336:
5332:
5315:
5312:
5309:
5306:
5303:
5297:
5288:
5284:
5281:
5274:
5267:
5248:
5244:
5240:
5235:
5231:
5225:
5222:
5216:
5208:
5204:
5200:
5195:
5191:
5185:
5182:
5176:
5168:
5164:
5160:
5155:
5151:
5145:
5142:
5132:
5129:
5125:
5123:
5122:Steiner point
5120:
5115:
5112:
5111:
5107:
5083:
5080:
5074:
5071:
5065:
5062:
5059:
5050:
5047:
5041:
5038:
5032:
5029:
5026:
5017:
5014:
5008:
5005:
4999:
4996:
4983:
4980:
4974:
4971:
4965:
4962:
4959:
4950:
4947:
4941:
4938:
4932:
4929:
4926:
4917:
4914:
4908:
4905:
4899:
4896:
4885:
4881:
4877:
4873:
4871:
4868:
4863:
4856:
4853:
4852:
4848:
4844:
4820:
4817:
4811:
4808:
4802:
4799:
4796:
4787:
4784:
4778:
4775:
4769:
4766:
4763:
4754:
4751:
4745:
4742:
4736:
4733:
4720:
4717:
4711:
4708:
4702:
4699:
4696:
4687:
4684:
4678:
4675:
4669:
4666:
4663:
4654:
4651:
4645:
4642:
4636:
4633:
4622:
4618:
4614:
4610:
4608:
4605:
4600:
4593:
4590:
4589:
4585:
4571:
4562:
4559:
4553:
4550:
4544:
4541:
4538:
4529:
4526:
4520:
4517:
4511:
4508:
4505:
4496:
4493:
4487:
4484:
4478:
4475:
4468:
4465:
4461:
4459:
4456:
4451:
4448:
4447:
4443:
4426:
4423:
4420:
4414:
4411:
4408:
4405:
4402:
4396:
4393:
4390:
4384:
4381:
4378:
4375:
4372:
4366:
4363:
4360:
4354:
4351:
4348:
4345:
4338:
4335:
4331:
4329:
4326:
4321:
4318:
4317:
4313:
4296:
4293:
4290:
4284:
4281:
4278:
4272:
4269:
4266:
4260:
4257:
4254:
4248:
4245:
4242:
4236:
4233:
4226:
4222:
4218:
4214:
4212:
4209:
4204:
4201:
4200:
4196:
4193:point of the
4192:
4189:
4172:
4169:
4166:
4163:
4160:
4154:
4148:
4145:
4142:
4139:
4136:
4130:
4124:
4121:
4118:
4115:
4112:
4102:
4099:
4095:
4093:
4090:
4085:
4082:
4081:
4077:
4061:
4057:
4054:
4051:
4048:
4045:
4039:
4034:
4030:
4027:
4024:
4021:
4018:
4012:
4007:
4003:
4000:
3997:
3994:
3991:
3981:
3977:
3973:
3969:
3967:
3964:
3959:
3956:
3955:
3951:
3934:
3931:
3928:
3925:
3922:
3917:
3914:
3908:
3902:
3899:
3896:
3893:
3890:
3885:
3882:
3876:
3870:
3867:
3864:
3861:
3858:
3853:
3850:
3840:
3836:
3832:
3828:
3826:
3823:
3818:
3815:
3814:
3810:
3796:
3793:
3790:
3787:
3784:
3777:
3774:
3770:
3768:
3765:
3760:
3757:
3756:
3752:
3735:
3732:
3729:
3723:
3720:
3717:
3711:
3708:
3705:
3699:
3696:
3693:
3687:
3684:
3681:
3675:
3672:
3665:
3662:
3658:
3656:
3653:
3648:
3645:
3644:
3640:
3636:
3622:
3619:
3616:
3613:
3610:
3607:
3604:
3601:
3598:
3595:
3592:
3585:
3582:
3578:
3576:
3573:
3568:
3565:
3564:
3560:
3556:
3552:
3538:
3535:
3532:
3529:
3526:
3523:
3520:
3517:
3514:
3511:
3508:
3501:
3498:
3494:
3492:
3489:
3484:
3481:
3480:
3476:
3472:
3468:
3464:
3450:
3447:
3444:
3441:
3438:
3435:
3432:
3429:
3422:
3419:
3415:
3413:
3410:
3405:
3402:
3401:
3397:
3393:
3389:
3375:
3372:
3369:
3366:
3363:
3356:
3353:
3349:
3347:
3344:
3339:
3336:
3335:
3331:
3328:
3326:Name; Symbol
3322:
3316:
3311:
3309:
3305:
3289:
3286:
3278:
3270:
3267:
3259:
3251:
3248:
3223:
3220:
3212:
3204:
3201:
3193:
3185:
3182:
3170:
3162:
3157:
3155:
3153:
3122:
3118:
3114:
3111:
3108:
3105:
3102:
3099:
3096:
3093:
3070:
3067:
3062:
3059:
3053:
3050:
3039:
3035:
3031:
3028:
3025:
3022:
3012:
3008:
3004:
3001:
2998:
2995:
2968:
2964:
2960:
2957:
2954:
2951:
2946:
2942:
2938:
2933:
2929:
2916:
2912:
2908:
2905:
2902:
2899:
2894:
2890:
2886:
2881:
2877:
2859:
2852:
2848:
2844:
2841:
2838:
2835:
2818:
2814:
2810:
2807:
2804:
2801:
2796:
2792:
2788:
2783:
2779:
2767:
2761:
2756:
2750:
2747:
2744:
2741:
2738:
2732:
2725:
2724:
2723:
2717:
2715:
2708:
2695:
2691:
2685:
2682:
2677:
2674:
2670:
2666:
2663:
2660:
2656:
2650:
2647:
2642:
2639:
2635:
2631:
2628:
2625:
2621:
2615:
2612:
2607:
2604:
2600:
2596:
2593:
2585:
2582:AA', BB', CC'
2578:
2571:
2564:
2545:
2535:
2533:
2508:
2505:
2502:
2499:
2496:
2490:
2487:
2485:
2472:
2468:
2464:
2459:
2455:
2451:
2446:
2442:
2435:
2432:
2430:
2417:
2413:
2409:
2404:
2400:
2396:
2391:
2387:
2380:
2377:
2375:
2367:
2364:
2361:
2358:
2355:
2349:
2318:
2315:
2312:
2309:
2306:
2300:
2295:
2291:
2287:
2285:
2265:
2261:
2257:
2252:
2248:
2244:
2239:
2235:
2228:
2216:
2212:
2208:
2206:
2189:
2181:
2178:
2172:
2167:
2159:
2156:
2150:
2145:
2137:
2134:
2121:
2118:
2115:
2113:
2105:
2102:
2099:
2096:
2093:
2090:
2087:
2084:
2078:
2049:
2043:
2039:
2035:
2030:
2026:
2022:
2017:
2013:
2008:
2004:
2001:
1997:
1993:
1990:
1987:
1984:
1981:
1977:
1973:
1964:
1951:
1943:
1939:
1935:
1930:
1926:
1922:
1917:
1913:
1906:
1903:
1895:
1891:
1887:
1882:
1878:
1874:
1869:
1865:
1858:
1855:
1847:
1843:
1839:
1834:
1830:
1826:
1821:
1817:
1810:
1802:
1799:
1789:
1784:
1782:
1780:
1774:
1770:
1766:
1759:
1755:
1751:
1745:
1741:
1735:
1731:
1725:
1721:
1715:
1709:
1705:
1696:
1688:
1682:
1678:
1672:
1668:
1662:
1658:
1653:
1648:
1643:
1642:
1641:
1640:
1639:
1626:
1621:
1617:
1613:
1610:
1607:
1604:
1599:
1595:
1591:
1586:
1582:
1577:
1572:
1568:
1564:
1561:
1558:
1555:
1550:
1546:
1542:
1537:
1533:
1528:
1523:
1519:
1515:
1512:
1509:
1506:
1501:
1497:
1493:
1488:
1484:
1467:
1463:
1449:
1444:
1440:
1436:
1431:
1427:
1423:
1418:
1414:
1409:
1404:
1400:
1396:
1391:
1387:
1383:
1378:
1374:
1369:
1364:
1360:
1356:
1351:
1347:
1343:
1338:
1334:
1324:
1319:
1312:
1305:
1298:
1291:
1284:
1280:
1279:
1278:
1277:
1276:
1273:
1264:
1262:
1259:
1242:
1239:
1236:
1233:
1230:
1227:
1223:
1220:
1217:
1214:
1211:
1208:
1204:
1201:
1198:
1195:
1192:
1189:
1165:
1148:
1144:
1122:
1118:
1114:
1110:
1106:
1101:
1097:
1093:
1089:
1085:
1080:
1076:
1072:
1068:
1059:
1052:
1034:
1029:
1008:
1006:
987:
984:
954:
951:
923:
920:
917:
914:
911:
905:
896:
878:
875:
872:
866:
863:
860:
857:
854:
846:
842:
818:
815:
809:
803:
800:
797:
794:
791:
783:
779:
770:
765:
763:
755:
749:
732:
729:
726:
723:
720:
714:
711:
705:
702:
699:
696:
693:
687:
684:
678:
675:
672:
669:
666:
660:
652:
621:
615:
612:
609:
606:
603:
597:
594:
588:
585:
582:
579:
576:
570:
562:
557:
533:
530:
527:
524:
521:
515:
510:
506:
502:
496:
493:
490:
487:
484:
481:
478:
475:
469:
462:Homogeneity:
461:
460:
459:
450:
442:
440:
424:
420:
396:
390:
382:
366:
344:
340:
316:
310:
302:
298:
297:
292:
286:
284:
280:
276:
275:Lemoine point
272:
268:
259:
257:
255:
254:
249:
244:
242:
238:
234:
230:
226:
222:
218:
214:
209:
207:
206:constructions
203:
199:
195:
191:
187:
183:
179:
175:
171:
167:
163:
151:
140:
137:(centered at
136:
123:
119:
105:
101:
88:
84:
71:
67:
63:
53:
41:
37:
33:
19:
9198:
9194:
9184:
9151:
9147:
9137:
9119:10.1142/7740
9109:
9071:
9064:
9054:
9050:
9037:
9016:
9004:. Retrieved
9000:
8990:
8978:. Retrieved
8974:
8964:
8939:
8935:
8925:
8913:
8901:. Retrieved
8897:
8887:
8875:. Retrieved
8871:
8858:
8846:. Retrieved
8842:
8829:
8804:
8800:
8791:
8782:
8776:. Retrieved
8763:
8750:
8746:
8742:
8737:
8697:Central line
8681:
8670:
8651:
8627:
8614:
8610:
8606:
8279:
8275:
8271:
8267:
8259:
7842:
7643:
7635:
7633:
7499:
7488:
7484:
7480:
7474:
7470:
7466:
7459:
7255:
7244:
7025:
7016:
7010:
6684:
6678:
6669:
6547:
6389:
6385:
6378:
6355:
6354:is called a
6349:
6243:
6237:
6222:
6218:
6214:
6204:
6203:is called a
6198:
6180:
6179:is called a
6174:
6164:
6162:
6106:
6010:
5938:
5854:
5794:
5698:
5644:
5569:
5564:early 1990s
5485:
5413:Exeter point
5404:
5338:
5278:
5127:
5113:
4879:
4875:
4861:
4854:
4616:
4612:
4598:
4591:
4463:
4458:Fermat point
4449:
4333:
4319:
4220:
4216:
4202:
4097:
4083:
3975:
3971:
3957:
3834:
3830:
3816:
3772:
3758:
3660:
3646:
3580:
3566:
3496:
3491:Circumcenter
3482:
3417:
3403:
3351:
3337:
3332:Description
3306:
3172:
3158:Non-examples
3152:Fermat point
3144:
2721:
2718:Fermat point
2709:
2586:
2576:
2569:
2562:
2543:
2539:
1965:
1803:
1797:
1793:
1790:Circumcenter
1772:
1768:
1764:
1757:
1753:
1749:
1743:
1739:
1733:
1729:
1723:
1719:
1713:
1707:
1703:
1700:
1686:
1680:
1676:
1670:
1666:
1660:
1656:
1646:
1476:
1465:
1322:
1310:
1303:
1296:
1289:
1282:
1281:The centers
1271:
1268:
1257:
1255:This region
1050:
1012:
897:
766:
761:
750:
636:
555:
553:and for all
446:
300:
294:
287:
267:Fermat point
263:
251:
245:
210:
190:circumcenter
169:
165:
159:
104:circumcircle
51:
36:
8671:Centers of
6679:A function
6466:since
5494:Parry point
4845:Centers of
4092:Mittenpunkt
3966:Nagel point
3575:Orthocenter
2551:and vertex
237:translation
217:equivariant
198:orthocenter
122:orthocenter
9284:Paul Yiu,
9208:2004.01677
9057:(1): 1–35.
9028:1608.08190
8778:2009-05-23
8722:Euler line
8673:tetrahedra
7017:normalized
229:reflection
150:Euler line
9273:Ed Pegg,
9233:214795185
9225:1420-9012
9201:(2): 81.
9176:122725235
9168:1422-6383
9091:663096629
8956:0002-9890
8677:simplices
8642:homothecy
8638:homothety
8352:α
8338:β
8324:γ
8304:α
8298:β
8292:γ
8234:
8225:
8213:
8197:
8191:−
8185:
8169:
8163:−
8157:
8141:∡
8126:
8120:−
8114:
8098:
8089:
8077:
8061:
8055:−
8049:
8033:∡
8018:
8012:−
8006:
7990:
7984:−
7978:
7962:
7953:
7941:
7925:∡
7905:
7889:
7873:
7861:△
7811:∡
7800:∡
7796:if either
7787:
7781:−
7775:
7754:∡
7741:
7732:
7720:
7702:△
7689:
7419:otherwise
7412:β
7355:γ
7298:α
7187:−
6812:→
6722:−
6645:otherwise
6622:≥
6608:≥
6591:−
6544:Excenters
6075:
6057:
6033:
5982:−
5913:−
5824:
5772:−
5757:
5739:
5721:
5667:
5613:
5592:
5537:−
5524:−
5453:−
5379:
5367:
5241:−
5201:−
5161:−
5081:π
5066:
5048:π
5033:
5015:π
5000:
4981:π
4975:−
4966:
4948:π
4942:−
4933:
4915:π
4909:−
4900:
4847:inversion
4818:π
4812:−
4803:
4785:π
4779:−
4770:
4752:π
4746:−
4737:
4718:π
4703:
4685:π
4670:
4652:π
4637:
4560:π
4545:
4527:π
4512:
4494:π
4479:
4424:−
4415:
4409:−
4394:−
4385:
4379:−
4364:−
4355:
4349:−
4191:Symmedian
4170:−
4146:−
4122:−
4055:−
4028:−
4001:−
3932:−
3900:−
3868:−
3733:−
3724:
3709:−
3700:
3685:−
3676:
3639:altitudes
3620:
3608:
3596:
3536:
3524:
3512:
3115:π
3090:⟺
3068:π
3054:
3032:π
3005:π
2981:⟺
2845:π
2827:⟺
2683:π
2667:
2648:π
2632:
2613:π
2597:
2465:−
2410:−
2258:−
2173:−
2036:−
1936:−
1888:−
1840:−
1424:≤
1384:≤
1344:≤
1231:≤
1212:≤
1193:≤
769:bijective
762:cyclicity
213:invariant
118:Altitudes
9300:Category
8690:See also
8684:centroid
8137:if
8029:if
7921:if
7750:if
7698:if
7473: :
7469: :
7362:if
7305:if
7023:so that
6601:if
6226:, where
3412:Centroid
3346:Incenter
2992:if
2873:if
2832:if
2775:if
2559:and let
1785:Examples
939:be 0 if
381:centroid
233:dilation
225:rotation
219:) under
194:incenter
186:centroid
178:triangle
162:geometry
87:centroid
70:incenter
66:incircle
8848:17 June
8821:2690608
7251:α, β, γ
7247:a, b, c
7245:Assume
7155:algebra
7100:for all
7021:a, b, c
7015:can be
6751:for all
6393:) then
6364:a, b, c
6336:
6307:
6189:a, b, c
3467:medians
2712:a, b, c
1652:scalene
1182:where
1180:
1151:
1139:a, b, c
1047:
1015:
1003:
974:
970:
941:
893:a, b, c
758:a, b, c
647:a, b, c
456:a, b, c
260:History
246:For an
176:in the
146:H, G, O
83:Medians
9231:
9223:
9174:
9166:
9125:
9089:
9079:
9006:25 May
8980:25 May
8954:
8903:1 July
8877:25 May
8819:
8753:≤ 2π/3
8640:, and
8560:
8552:
8525:
8517:
8446:
8438:
8411:
8403:
8349:
8341:
8335:
8327:
7895:
7879:
7117:
6305:where
3475:lamina
3282:
3276:
3263:
3257:
3216:
3210:
3197:
3191:
1326:where
1147:domain
1060:, are
558:> 0
359:where
281:, and
133:
131:
116:
114:
98:
96:
81:
79:
60:
58:
46:
9288:from
9277:from
9267:from
9229:S2CID
9203:arXiv
9172:S2CID
9047:(PDF)
9023:arXiv
8817:JSTOR
8729:Notes
7487:<
7483:<
7460:Then
6147:1992
6101:1978
6005:1961
5933:1987
5789:1985
5693:1987
5639:1989
5480:1986
5399:1985
3145:Then
2570:ABC'
1684:from
235:, or
182:plane
174:point
172:is a
9221:ISSN
9164:ISSN
9123:ISBN
9087:OCLC
9077:ISBN
9008:2009
8982:2009
8952:ISSN
8905:2009
8879:2009
8850:2022
7383:>
7369:>
7326:<
7312:<
7162:and
7108:>
6687:if
6548:Let
6379:Let
6358:if
3109:>
3026:>
2999:>
2939:>
2887:>
2839:>
2789:>
2722:Let
2566:and
2563:AB'C
2544:A'BC
2540:Let
1966:Let
1592:>
1543:>
1494:>
972:and
834:and
653:are
196:and
164:, a
102:and
64:and
9213:doi
9156:doi
9115:doi
8944:doi
8809:doi
8367:If
8316:is
8231:sec
8222:sec
8210:cos
8194:sec
8182:cos
8166:sec
8154:cos
8123:sec
8111:cos
8095:sec
8086:sec
8074:cos
8058:sec
8046:cos
8015:sec
8003:cos
7987:sec
7975:cos
7959:sec
7950:sec
7938:cos
7902:cos
7886:cos
7870:cos
7784:sec
7772:cos
7738:sec
7729:sec
7717:cos
7686:cos
7500:If
7028:= 0
6683:is
6213:△,
6110:360
6072:cos
6054:cos
6030:cos
6014:356
5942:192
5858:181
5815:sec
5798:179
5754:cos
5736:cos
5718:sec
5702:175
5664:sec
5648:174
5610:sec
5589:tan
5573:173
5489:111
5376:cos
5364:cos
5117:99
5063:sec
5030:sec
4997:sec
4963:sec
4930:sec
4897:sec
4800:sin
4767:sin
4734:sin
4700:sin
4667:sin
4634:sin
4542:csc
4509:csc
4476:csc
4412:cos
4382:cos
4352:cos
3721:cos
3697:cos
3673:cos
3617:sec
3605:sec
3593:sec
3533:cos
3521:cos
3509:cos
3469:.
3051:csc
2664:csc
2629:csc
2594:csc
2577:ABC
2528:so
1798:ABC
1137:so
1054:365
756:of
412:or
332:or
293:'s
208:.
180:'s
168:or
160:In
52:ABC
9302::
9263:,
9227:.
9219:.
9211:.
9199:76
9197:.
9193:.
9170:.
9162:.
9152:55
9150:.
9146:.
9121:.
9099:^
9085:.
9053:.
9049:.
8999:.
8973:.
8950:.
8940:85
8938:.
8934:.
8896:.
8870:.
8866:.
8841:.
8837:.
8815:.
8805:67
8803:.
8781:.
8771:.
8644:.
8636:,
8632:,
8613:,
8609:,
8278:,
8274:,
7506:af
6388:=
6366:.
6221:,
6217:,
6191:.
6167:.
5408:22
5342:21
4865:18
4858:17
4602:16
4595:15
4453:13
4323:11
4206:10
3641:.
3561:.
3477:.
3398:.
3154:.
2557:BC
2553:A'
2549:BC
1781:.
1771:,
1767:,
1756:=
1752:=
1742:=
1737:,
1732:=
1727:,
1722:=
1706:⊆
1679:=
1674:,
1669:=
1664:,
1659:=
1469:13
1314:40
1309:,
1307:24
1302:,
1300:22
1295:,
1288:,
895:.
764:.
447:A
439:.
277:,
273:,
269:,
256:.
231:,
227:,
192:,
188:,
9292:.
9281:.
9235:.
9215::
9205::
9178:.
9158::
9131:.
9117::
9093:.
9055:6
9031:.
9025::
9010:.
8984:.
8958:.
8946::
8907:.
8881:.
8852:.
8823:.
8811::
8751:C
8749:,
8747:B
8745:,
8743:A
8617:)
8615:a
8611:b
8607:c
8605:(
8601:f
8587:.
8584:)
8581:b
8578:,
8575:c
8572:,
8569:a
8566:(
8563:f
8556:|
8549:)
8546:c
8543:,
8540:a
8537:,
8534:b
8531:(
8528:f
8521:|
8514:)
8511:a
8508:,
8505:b
8502:,
8499:c
8496:(
8493:f
8473:,
8470:)
8467:c
8464:,
8461:b
8458:,
8455:a
8452:(
8449:f
8442:|
8435:)
8432:a
8429:,
8426:c
8423:,
8420:b
8417:(
8414:f
8407:|
8400:)
8397:b
8394:,
8391:a
8388:,
8385:c
8382:(
8379:f
8369:f
8355:.
8345:|
8331:|
8301::
8295::
8282:)
8280:a
8276:b
8272:c
8270:(
8237:B
8228:A
8219:+
8216:C
8205::
8200:B
8188:B
8177::
8172:A
8160:A
8144:C
8129:C
8117:C
8106::
8101:A
8092:C
8083:+
8080:B
8069::
8064:A
8052:A
8036:B
8021:C
8009:C
7998::
7993:B
7981:B
7970::
7965:C
7956:B
7947:+
7944:A
7928:A
7908:C
7898::
7892:B
7882::
7876:A
7822:.
7814:C
7803:B
7790:A
7778:A
7765:,
7757:A
7744:C
7735:B
7726:+
7723:A
7710:,
7692:A
7680:{
7675:=
7672:)
7669:c
7666:,
7663:b
7660:,
7657:a
7654:(
7651:f
7639:3
7636:X
7624:f
7606:.
7603:)
7600:b
7597:,
7594:a
7591:,
7588:c
7585:(
7582:f
7578:c
7575::
7572:)
7569:a
7566:,
7563:c
7560:,
7557:b
7554:(
7551:f
7547:b
7544::
7541:)
7538:c
7535:,
7532:b
7529:,
7526:a
7523:(
7520:f
7516:a
7502:f
7489:c
7485:b
7481:a
7475:γ
7471:β
7467:α
7462:f
7439:.
7436:)
7428:a
7425:(
7405:,
7402:)
7394:a
7391:(
7386:c
7380:a
7372:b
7366:a
7348:,
7345:)
7337:a
7334:(
7329:c
7323:a
7315:b
7309:a
7292:{
7287:=
7284:)
7281:c
7278:,
7275:b
7272:,
7269:a
7266:(
7263:f
7226:.
7223:f
7218:3
7214:)
7210:c
7207:+
7204:b
7201:+
7198:a
7195:(
7190:1
7183:)
7179:c
7176:b
7173:a
7170:(
7160:f
7141:.
7138:)
7135:c
7132:,
7129:b
7126:,
7123:a
7120:(
7114:,
7111:0
7105:t
7095:)
7092:c
7089:,
7086:b
7083:,
7080:a
7077:(
7074:f
7071:=
7068:)
7065:c
7062:t
7059:,
7056:b
7053:t
7050:,
7047:a
7044:t
7041:(
7038:f
7026:n
7013:f
6991:.
6988:)
6985:b
6982:,
6979:a
6976:,
6973:c
6970:(
6967:f
6964::
6961:)
6958:a
6955:,
6952:c
6949:,
6946:b
6943:(
6940:f
6937::
6934:)
6931:c
6928:,
6925:b
6922:,
6919:a
6916:(
6913:f
6893:)
6890:b
6887:,
6884:a
6881:,
6878:c
6875:(
6872:f
6868:)
6865:a
6862:,
6859:c
6856:,
6853:b
6850:(
6847:f
6841:2
6837:)
6833:c
6830:,
6827:b
6824:,
6821:a
6818:(
6815:f
6809:)
6806:c
6803:,
6800:b
6797:,
6794:a
6791:(
6771:.
6768:c
6765:,
6762:b
6759:,
6756:a
6746:)
6743:b
6740:,
6737:c
6734:,
6731:a
6728:(
6725:f
6719:=
6716:)
6713:c
6710:,
6707:b
6704:,
6701:a
6698:(
6695:f
6681:f
6649:.
6638:1
6628:,
6625:c
6619:a
6611:b
6605:a
6594:1
6585:{
6580:=
6577:)
6574:c
6571:,
6568:b
6565:,
6562:a
6559:(
6556:f
6513:)
6510:a
6507:,
6504:c
6501:,
6498:b
6495:(
6492:f
6489:=
6479:)
6476:b
6473:=
6470:a
6462:(
6457:)
6454:c
6451:,
6448:a
6445:,
6442:b
6439:(
6436:f
6433:=
6426:)
6423:c
6420:,
6417:b
6414:,
6411:a
6408:(
6405:f
6390:b
6386:a
6381:f
6360:P
6352:P
6340:X
6324:)
6321:X
6318:(
6315:f
6293:)
6290:C
6287:(
6284:f
6281::
6278:)
6275:B
6272:(
6269:f
6266::
6263:)
6260:A
6257:(
6254:f
6240:P
6228:△
6223:c
6219:b
6215:a
6209:P
6201:P
6185:P
6177:P
6131:a
6128:A
6107:X
6084:3
6081:C
6066:3
6063:B
6051:2
6048:+
6042:3
6039:A
6011:X
5991:)
5988:c
5985:b
5979:b
5976:a
5973:+
5970:a
5967:c
5964:(
5961:c
5958:b
5939:X
5916:a
5910:c
5907:+
5904:b
5897:2
5893:)
5889:c
5886:+
5883:b
5880:(
5877:a
5855:X
5833:4
5830:A
5819:4
5795:X
5775:1
5766:2
5763:C
5748:2
5745:B
5730:2
5727:A
5699:X
5676:2
5673:A
5645:X
5622:2
5619:A
5607:+
5601:2
5598:A
5570:X
5545:2
5541:c
5532:2
5528:b
5519:2
5515:a
5511:2
5507:a
5486:X
5466:)
5461:4
5457:a
5448:4
5444:c
5440:+
5435:4
5431:b
5427:(
5424:a
5405:X
5382:C
5373:+
5370:B
5360:1
5339:X
5319:)
5316:c
5313:,
5310:b
5307:,
5304:a
5301:(
5298:f
5249:2
5245:b
5236:2
5232:a
5226:b
5223:a
5217::
5209:2
5205:a
5196:2
5192:c
5186:a
5183:c
5177::
5169:2
5165:c
5156:2
5152:b
5146:c
5143:b
5128:S
5114:X
5090:)
5084:3
5075:+
5072:C
5069:(
5060::
5057:)
5051:3
5042:+
5039:B
5036:(
5027::
5024:)
5018:3
5009:+
5006:A
5003:(
4990:)
4984:3
4972:C
4969:(
4960::
4957:)
4951:3
4939:B
4936:(
4927::
4924:)
4918:3
4906:A
4903:(
4882:′
4880:N
4876:N
4862:X
4855:X
4827:)
4821:3
4809:C
4806:(
4797::
4794:)
4788:3
4776:B
4773:(
4764::
4761:)
4755:3
4743:A
4740:(
4727:)
4721:3
4712:+
4709:C
4706:(
4697::
4694:)
4688:3
4679:+
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4506::
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3715:)
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2002:=
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1974:f
1952:.
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