X6

SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT)

X6 is the point of concurrence of the symmedians (i.e., reflections of medians in corresponding angle bisectors). X6 is the point which, when given by actual trilinear distances x,y,z, minimizes \(x^{2}\) + \(y^{2}\) + \(z^{2}\).
< A B C >: Reference Triangle △ABC < Tan_a Tan_b Tan_c >: Tangential Triangle △Tan_a Tan_b Tan_c [ A B ]: Sideline AB (Base c = 1) [ B C ]: Sideline BC (Side a) [ C A ]: Sideline CA (Side b) circumcircle: Circumcircle of △ABC [ A Tan_a ]: A-Symmedian [ B Tan_b ]: B-Symmedian [ C Tan_c ]: C-Symmedian { 1, 9 }: Incenter–Symmedian Line { 3, 15 }: Brocard Axis { 13, 14 }: Central Line L(13, 14) { 17, 18 }: Central Line L(17, 18) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_3 = CIRCUMCENTER (0.5000, 0.1021) Tan_a = Tangential Vertex Tan_a (1.1000, 0.4899) Tan_b = Tangential Vertex Tan_b (-0.1579, 0.7735) Tan_c = Tangential Vertex Tan_c (0.5000, -2.4495) X_1 = INCENTER (0.5714, 0.2333) X_9 = MITTENPUNKT (0.5110, 0.1615) X_15 = 1st ISODYNAMIC POINT (0.5519, 0.1878) X_13 = 1st ISOGONIC CENTER (FERMAT POINT, TORRICELLI POINT) (0.5879, 0.2819) X_14 = 2nd ISOGONIC CENTER (0.9134, -0.1143) X_17 = 1st NAPOLEON POINT (0.5666, 0.2383) X_18 = 2nd NAPOLEON POINT (0.4434, 0.1317) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6000, 0.2672) A B C X3 Tana Tanb Tanc X1 X9 X15 X13 X14 X17 X18 X6

X6 (SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT)) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

  • Derived Triangle: Tangential Triangle △TanaTanbTanc — vertices Tana, Tanb, Tanc (Bounded by tangents to the circumcircle at A, B, C)

  • Circle: Circumcircle (circumcircle) — Passes through vertices A, B, C; centered at X3 CIRCUMCENTER

  • Concurrence Lines: A–Tana, B–Tanb, C–Tanc concurring at X6

  • Related Centers: X3 (CIRCUMCENTER)

Trilinears: \(a : b : c\)
Barycentrics: \(a^{2} : b^{2} : c^{2}\)
Tripolars: \(b c \sqrt{2(b^{2} + c^{2}) - a^{2}} : :\)
Search Value (6, 9, 13): \(0.992908495066\)
▸ Alternate Coordinates7 forms

  • Trilinears: \(\sin A : \sin B : \sin C\)

  • Trilinears: \((1 + \cos A)(1 - \cos A + \cos B + \cos C) : :\)

  • Trilinears: \(\cot B/2 + \cot C/2 : :\)

  • Barycentrics: \(S_{B} + S_{C} : :\)

  • Barycentrics: \(S_{A} - S_{\omega} : :\)

  • Barycentrics: \(\cot A - \cot \omega : :\)

  • Barycentrics: \(\cot B + \cot C - \cot A + \cot \omega : :\)

▸ Linear Combinations5 combinations

  • X6 = \((r^{2} + s^{2} + 2rR)\) X1 - \(6rR\) X2 - \(2r^{2}\) X3

  • X6 = \((1 + \sqrt{3} \tan(\omega ))\) X13 + \((1 - \sqrt{3} \tan(\omega ))\) X14

  • X6 = \((1 + \sqrt{3} \tan(\omega ))\) X15 + \((1 - \sqrt{3} \tan(\omega ))\) X16

  • X6 = \((3 + 5 \sqrt{3} \tan(\omega ))\) X17 + \((3 - 5 \sqrt{3} \tan(\omega ))\) X18

  • (The above four combos for X6 found by Peter Moses, November, 2011)

X6 lies on the Walsmith rectangular hyperbola, the Thomson cubic.

Related Collection 1 Topic Guides

T_1: Isogonal Conjugate • T_5: Circle Inversion • T_8: The Brocard Axis • T_12: The Fuhrmann & Spieker Central Lines • T_13: The Isogonic & Napoleon Central Lines • T_15: The Circumcircle • T_18: The Apollonian Circles • T_19: The Brocard & Orthocentroidal Circles • T_21: Orthic & Tangential Triangles • T_23: Incentral & Excentral Triangles

▸ Incident Central Lines127 lines

{1, 9}, {2, 69}, {3, 15}, {4, 53}, {5, 68}, {7, 294}, {8, 594}, {10, 1377}, {13, 14}, {17, 18}, {19, 34}, {21, 941}, {22, 251}, {23, 353}, {24, 54}, {25, 51}, {26, 143}, {27, 1246}, {31, 42}, {33, 204}, {36, 609}, {40, 380}, {41, 48}, {43, 87}, {57, 222}, {60, 1169}, {64, 185}, {66, 427}, {67, 125}, {70, 1594}, {74, 112}, {75, 239}, {76, 83}, {77, 241}, {88, 89}, {98, 262}, {99, 729}, {100, 739}, {101, 106}, {105, 1002}, {110, 111}, {145, 346}, {157, 248}, {160, 237}, {162, 1013}, {169, 942}, {181, 197}, {190, 192}, {194, 384}, {210, 612}, {226, 1751}, {256, 1580}, {264, 287}, {274, 1218}, {279, 1170}, {281, 1146}, {282, 1256}, {291, 985}, {292, 869}, {297, 317}, {305, 1241}, {314, 981}, {330, 1258}, {344, 1332}, {354, 374}, {442, 1714}, {493, 1583}, {494, 1584}, {513, 1024}, {517, 998}, {519, 996}, {523, 879}, {560, 1631}, {561, 720}, {588, 1599}, {589, 1600}, {593, 1171}, {595, 1126}, {598, 671}, {603, 1035}, {644, 1120}, {657, 1459}, {662, 757}, {688, 882}, {689, 703}, {691, 843}, {692, 1438}, {694, 1084}, {706, 1502}, {717, 789}, {750, 899}, {753, 825}, {755, 827}, {840, 919}, {846, 1051}, {893, 1403}, {909, 1415}, {911, 1461}, {939, 1802}, {943, 1612}, {947, 1622}, {959, 961}, {963, 1208}, {967, 1790}, {971, 990}, {986, 1046}, {1073, 3343}, {1096, 1859}, {1112, 1177}, {1131, 1132}, {1139, 1140}, {1166, 1601}, {1173, 1614}, {1174, 1617}, {1195, 1399}, {1201, 1696}, {1214, 1708}, {1327, 1328}, {1362, 1416}, {1398, 1425}, {1423, 1429}, {1718, 1781}, {1826, 1837}, {1836, 1839}, {1854, 1858}, {3342, 3351}, {3344, 3350}

▸ Functional Relations134 relations

Notes

X6 in Navigation: A talk about the symmedian point, by William Lionheart.

X6 and other triangle centers play a fundamental part in Yuri I. Loginov’s “Energy methods for single-position passive radar based on special points of a triangle”, downloadable in Russian (EnergyMethodsRussian.pdf) or as an English translation (EnergyMethodsEnglish.pdf).

Let A'B'C' be the pedal triangle of an arbitrary point X, and let S(X) be the vector sum XA' + XB' + XC'. Then

S(X) = (0 vector) if and only if X = X6.

The “if” implication is equivalent to the well known fact that X6 is the centroid of its pedal triangle, and the converse was proved by Barry Wolk (Hyacinthos #19, Dec. 23, 1999).

X6 is the radical trace of the 1st and 2nd Lemoine circles. (Peter J. C. Moses, 8/24/03)

X6 is the perspector of ABC and the medial triangle of the orthic triangle of ABC. (Randy Hutson, 8/23/2011)

Ross Honsberger, Episodes in Nineteenth and Twentieth Century Euclidean Geometry, Mathematical Association of America, 995. Chapter 7: The Symmedian Point.

Let A' be the center of the conic through the contact points of the incircle and the A-excircle with the sidelines of ABC. Define B' and C' cyclically. Let A″ be the center of the conic through the contact points of the B- and C- excircles with the sidelines of ABC. Define B″ and C″ cyclically. The triangles A'B'C' and A″B″C″ are perspective at X6. See also X25, X218, X222, X940, X1743. (Randy Hutson, July 23, 2015)

The tangents at A, B, C to the Thomson cubic K002 concur in X6. Let Ha be the foot of the A-altitude. Let Ba, Ca be the feet of perpendiculars from Ha to CA, AB, resp. Let A' be the orthocenter of HaBaCa, and define B' and C' cyclically. The lines AA', BB', CC' concur in X6. (Randy Hutson, November 18, 2015)

Let Ja, Jb, Jc be the excenters and I the incenter of ABC. Let Ka be the symmedian point of JbJcI, and define Kb and Kc cyclically. Then KaKbKc is perspective to JaJbJc at X6. (Randy Hutson, February 10, 2016)

Let A'B'C' be the triangle described in ADGEOM #2697 (8/26/2015, Tran Quang Hung); then X6 = X6467-of-A'B'C'. (Randy Hutson, June 27, 2018)

X6 is the perspector of every pair of these triangles: anticevian triangle of X3, submedial triangle, unary cofactor triangle of submedial triangle, unary cofactor triangle of the intangents triangle, unary cofactor triangle of the extangents triangle. (Randy Hutson, June 27, 2018)

Let A'B'C' be the tangential triangle of the Jerabek hyperbola. Let A″ be the pole wrt circumcircle of line B'C', and define B″, C″ cyclically. The lines A’″, B'B″, C'C″ concur in X6. (Randy Hutson, November 30, 2018)

Let A'B'C' be the half-altitude (midheight) triangle. Let LA be the line through A parallel to B'C', and define LB and LC cyclically. Let A″ = LB∩LC, and define B″, C″ cyclically. The lines A’″, B'B″, C'C″ concur in X6. (Randy Hutson, November 30, 2018)

X6 is the unique point that is the centroid of its pedal triangle. (Randy Hutson, June 7, 2019)

Let A'B'C' be any one of {Lucas(t) central triangle, Lucas(t) tangents triangle, Lucas(t) inner triangle} (for arbitrary t). Let A″ be the trilinear pole of line B'C', and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X6. (Randy Hutson, July 11, 2019)

Let MaMbMc be the medial triangle, and HaHbHc the orthic triangle. Let A* be the centroid of AMaHa, and define B* and C* cyclically. A*B*C* is inversely similar to ABC, and the lines MaA*, MbB*, McC* concur in X6. (Randy Hutson, July 11, 2019)

X6 is the intersection of the isotomic conjugate of the polar conjugate of the Euler line (i.e., line X2X6), and the polar conjugate of the isotomic conjugate of the Euler line (i.e., line X4X6). (Randy Hutson, July 11, 2019)

X6 is the pole of the Euler line wrt each conic passing through each of the following sets of four points: {X13, X14, X15, X16}, {X13, X14, X17, X18}, {X13, X14, X61, X62}, {X15, X16, X17, X18}, {X17, X18, X61, X62}. (Randy Hutson, January 17, 2020)

Let OA be the circle centered at the A-vertex of the 2nd Brocard triangle and passing through A; define OB and OC cyclically. X6 is the radical center of OA, OB, OC. (Randy Hutson, August 30, 2020)

In the plane of a triangle ABC, let

  • Oa = circle with diameter BC, and define Ob and Oc cyclically;

  • A' = apex of equilateral triangle on side BC, and define B' and C' cyclically;

  • Ab = BA'∩Oa, and define Bc and Ca cyclically;

  • Ac = CA'∩Oa, and define Ba and Cb cyclically;

  • A″= BcBa∩CaCb, and define B″ and C″ cyclically.

The triangle A″B″C″ is perspective to ABC, and the perspector is X6. (Dasari Naga Vijay Krishna, April 19, 2021)

Let P and Q be distinct points in a plane, and let T be a triangle rigidly rotating about P. Let X be a triangle center, and let

  • XT = X-of-T;

  • O = a circle with center Q';

  • {A, B, C} = vertices of T;

  • A' = inverse of A in O, and define B' and C' cyclically;

  • T' = A'B'C';

  • X'T = X-of-T'.

    1. If X = Xk for k = 3, 6, 15, 16, 61, 62, then the locus of X'T is a conic.

    1. If X = X15 or X = X16, the conic is a circle.

    1. If P is one of the points Xk, for k = 3, 6, 15, 16, 61, 62, then the locus of X'T is a single point, and the points O, XT, X'T are collinear. For example, if P = X6, then the locus of X6-of-T' is collinear with O and X6-of-T.

  • Videos:

  • Inversive Image of Pivoting Triangle, Part II: Conic Loci of X3 and X6 of Inversive (Dan Reznik, August 15, 2021)

Let Γ denote the circumcircle. The trilinear polar of every point on Γ passes through X6; conversely, the trilinear pole of every line through X6 lies on Γ. The trilinear polar of every point on the Lemoine axis (the polar of X6 with respect to Γ) is a line tangent to the Brocard inellipse; conversely, the trilinear pole of every line tangent to the Brocard inellipse lies on the Lemoine axis. (Dan Reznik, February 3, 2023)

Let P be a point in the plane of T = ABC, and let T'=A'B'C' be the cevian triangle of P. Let A″ = BC∩B'C', and define B″ and C″ cycllically. By Desargues’ theorem, the points A″,B″,C″ lie on the perspectrix of the ktriangles T and T'. Let Ka be the circle that passes through the points A,A',A″, and define Kb and Kc cyclically. As is well known, if P=X1, then the circles Ka,Kb,Kc meet in X15 and X16, and their radical axis is the Brocard axis. We offer the following new observation. The locus of P such that Ka,Kb,Kc intersect in exactly two points, Z1 and Z2, is the curve Q066, which is the Stammler quartic (the isogonal image of the Stammler hyperbola). The curve Q066 passes through the excenters and the triangle centers Xk for k=1, 2, 4, 254, and others, listed in Q066, Stammler quartic) . Moreover, the line Z1Z2 passes through the point X6-of-T. See the figures in Q066 and line Z1Z2 . (Bernard Gibert and Dan Reznik, February 12, 2023)

X6 is the {X15,X16}-harmonic conjugate of X3.