X(6) = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT)¶
Trilinears
\(a : b : c\)
\(sin A : sin B : sin C\)
\((1 + cos A)(1 - cos A + cos B + cos C) : :\)
\(cot B/2 + cot C/2 : :\)
Barycentrics
\(a2 : b2 : c2\)
\(SB + SC : :\)
\(SA - SW : :\)
\(cot A - cot ω : :\)
\(cot B + cot C - cot A + cot ω : :\)
Notes
X(6) is the point of concurrence of the symmedians (i.e., reflections of medians in corresponding angle bisectors). X(6) is the point which, when given by actual trilinear distances x,y,z, minimizes x2 + y2 + z2.
X(6) in Navigation: A talk about the symmedian point, by William Lionheart.
X(6) 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 or as an English translation.
If you have The Geometer’s Sketchpad, you can view Symmedian point. If you have GeoGebra, you can view Symmedian point.
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 = X(6).
The “if” implication is equivalent to the well known fact that X(6) is the centroid of its pedal triangle, and the converse was proved by Barry Wolk (Hyacinthos #19, Dec. 23, 1999).
X(6) is the radical trace of the 1st and 2nd Lemoine circles. (Peter J. C. Moses, 8/24/03)
X(6) 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 X(6). See also X(25), X(218), X(222), X(940), X(1743). (Randy Hutson, July 23, 2015)
The tangents at A,B,C to the Thomson cubic K002 concur in X(6). 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 X(6). (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 X(6). (Randy Hutson, February 10, 2016)
Let A’B’C’ be the triangle described in ADGEOM #2697 (8/26/2015, Tran Quang Hung); then
X(6) = X(6467)-of-A’B’C’. (Randy Hutson, June 27, 2018)
X(6) is the perspector of every pair of these triangles: anticevian triangle of X(3), 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 X(6). (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 X(6). (Randy Hutson, November 30, 2018)
X(6) 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 A″, BB″, CC″ concur in X(6). (Randy Hutson, July 11, 2019)
Let A’B’C’ be the medial triangle, and A″B″C″ the orthic triangle. Let A* be the centroid of AA’A″, and define B* and C* cyclically. A*B*C* is inversely similar to ABC, and the lines A’A*, B’B*, C’C* concur in X(6). (Randy Hutson, July 11, 2019)
X(6) is the intersection of the isotomic conjugate of the polar conjugate of the Euler line (i.e., line X(2)X(6)), and the polar conjugate of the isotomic conjugate of the Euler line (i.e., line X(4)X(6)). (Randy Hutson, July 11, 2019)
X(6) is the pole of the Euler line wrt each conic passing through each of the following sets of four points: {X(13), X(14), X(15), X(16)}, {X(13), X(14), X(17), X(18)}, {X(13), X(14), X(61), X(62)}, {X(15), X(16), X(17), X(18)}, {X(17), X(18), X(61), X(62)}. (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. X(6) 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 X(6). (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 = X(k) for k = 3, 6, 15, 16, 61, 62, then the locus of X’T is a conic. (2) If X = X(15) or X = X(16), the conic is a circle. (3) If P is one of the points X(k), 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 = X(6), then the locus of X(6)-of-T’ is collinear with O and X(6)-of-T. Videos: Surprisingly Stationary Symmedian Point of the Inversive Image of an X(6)-Pivoting Triangle Inversive Image of Pivoting Triangle, Part I: Stationary Symmedian Point X(6) of Inversive Inversive Image of Pivoting Triangle, Part II: Conic Loci of X(3) and X(6) of Inversive (Dan Reznik, August 15, 2021)
Let Γ denote the circumcircle. The trilinear polar of every point on Γ passes through X(6); conversely, the trilinear pole of every line through X(6) lies on Γ. The trilinear polar of every point on the Lemoine axis (the polar of X(6) 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=:ref:X(1) <X(1)>, then the circles Ka,Kb,Kc meet in X(15) and X(16), 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 X(k) for k=1, 2, 4, 254, and others, listed in Q066, Stammler quartic) . Moreover, the line Z1Z2 passes through the point X(6)-of-T. See the figures in Q066 and line Z1Z2 . (Bernard Gibert and Dan Reznik, February 12, 2023)
X(6) lies on the Walsmith rectangular hyperbola, the Thomson cubic, and these 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
X(6) is the {X(15),:ref:X(16) <X(16)>}-harmonic conjugate of X(3). For a list of other harmonic conjugates of X(6), click Tables at the top of this page.
X(6) = midpoint of X(i) and X(j) for these (i,j): (32,5028), (39,5052), (69,193), (125,5095), (187,5107), (1689, 1690)
X(6) = reflection of X(i) in X(j) for these (i,j): (1,1386), (2,597), (3,182), (67,125), (69,141), (159,206), (182,575), (592,2), (694,1084), (1350,3), (1351,576), (1352,5), (32113,468)
X(6) = isogonal conjugate of X(2)
X(6) = isotomic conjugate of X(76)
X(6) = cyclocevian conjugate of X(1031)
X(6) = circumcircle-inverse of X(187)
X(6) = orthocentroidal-circle-inverse of X(115)
X(6) = 1st-Lemoine-circle-inverse of X(1691)
X(6) = complement of X(69)
X(6) = anticomplement of X(141)
X(6) = anticomplementary conjugate of X(1369)
X(6) = complementary conjugate of X(1368)
X(6) = crossdifference of every pair of points on line X(30)X(511)
X(6) = insimilicenter of 1st and 2nd Kenmotu circles
X(6) = exsimilicenter of circumcircle and (1/2)-Moses circle
X(6) = harmonic center of circumcircle and Gallatly circle
X(6) = perspector of polar circle wrt Schroeter triangle
X(6) = X(i)-Ceva conjugate of X(j) for these (i,j): (1,55), (2,3), (3,154), (4,25), (7,1486), (8,197), (9,198), (10,199), (54,184), (57, 56), (58,31), (68,161), (69,159), (74,1495), (76,22), (81,1), (83,2), (88,36), (95,160), (98,237), (110,512), (111,187), (222,221), (249,110), (251,32), (264,157), (275,4), (284,48), (287,1503), (288,54), (323,399), (394,1498), (1613,3360)
X(6) = cevapoint of X(i) and X(j) for these (i,j): (1,43), (2,194), (31,41), (32,184), (42,213), (51,217)
X(6) = X(i)-cross conjugate of X(j) for these (i,j): (25,64), (31,56), (32,25), (39,2), (41,55), (42,1), (51,4), (184,3), (187,111), (213,31), (217,184), (237,98), (512,110)
X(6) = crosspoint of X(i) and X(j) for these (i,j): (1,57), (2,4), (9,282), (54,275), (58,81), (83, 251), (110,249), (266,289)
X(6) = crosssum of X(i) and X(j) for these (i,j): (1,9), (3,6), (4,1249), (5,216), (10,37), (11,650), (32,206), (39,141), (44,214), (56,478), (57,223), (114,230), (115,523), (125,647), (128,231), (132,232), (140,233), (142,1212), (188,236), (226,1214), (244,661), (395,619), (396, 618), (512,1084), (513,1015), (514,1086), (522,1146), (570,1209), (577,1147), (590,641), (615,642), (1125,1213), (1196,1368), (5408,5409)
X(6) = X(i)-Hirst inverse of X(j) for these (i,j): (1,238), (2,385), (3,511), (15,16), (25,232), (56,1458), (58,1326), (523,1316), (1423,1429)
X(6) = X(i)-line conjugate of X(j) for these (i,j): (1,518), (2,524), (3,511)
X(6) = X(i)-aleph conjugate of X(j) for these (i,j): (1,846), (81,6), (365,1045), (366,191), (509,1046)
X(6) = X(i)-beth conjugate of X(j) for these (i,j): (6,604), (9,9), (21,1001), (101,6), (284,6), (294,6), (644,6), (645,6), (651,6), (652,7), (666,6)
X(6) = exsimilicenter of circumcircle and (1/2)-Moses circle; the insimilicenter is X(5013)
X(6) = homothetic center of outer Napoleon triangle and pedal triangle of X(15)
X(6) = homothetic center of inner Napoleon triangle and pedal triangle of X(16)
X(6) = trilinear product of vertices of Thomson triangle
X(6) = orthocenter of X(i)X(j)X(k) for these (i,j,k): (2,4,1640), (3,4,879), (3,64,2435)
X(6) = intersection of tangents at X(3) and X(4) to Darboux cubic K004
X(6) = radical trace of circumcircle and Ehrmann circle
X(6) = one of two harmonic traces of Ehrmann circles; the other is X(23)
X(6) = X(3734)-of-1st anti-Brocard-triangle
X(6) = X(182)-of-anti-McCay triangle
X(6) = intersection of tangents to 2nd Brocard circle at PU(1) (i.e., pole of line X(39)X(512) wrt 2nd Brocard circle)
X(6) = intersection of diagonals of trapezoid PU(1)PU(39)
X(6) = intersection of diagonals of trapezoid PU(6)PU(33)
X(6) = intersection of diagonals of trapezoid PU(31)PU(33)
X(6) = the point in which the extended legs P(6)U(31) and U(6)P(31) of the trapezoid PU(6)PU(31) meet
X(6) = trilinear pole of PU(i) for these i: 2, 26
X(6) = trilinear product of PU(8)
X(6) = barycentric product of PU(i) for these i: 1, 17, 113, 114, 115, 118, 119
X(6) = crossdifference of PU(i) for these i: 24, 41
X(6) = midpoint of PU(i) for these i: 45, 46, 54
X(6) = bicentric sum of PU(i) for these i: 45, 46, 54, 62
X(6) = Zosma transform of X(19)
X(6) = trilinear square of X(365)
X(6) = radical center of {circumcircle, Parry circle, Parry isodynamic circle}; see X(2)
X(6) = PU(62)-harmonic conjugate of X(351)
X(6) = vertex conjugate of PU(118)
X(6) = eigencenter of orthocentroidal triangle
X(6) = eigencenter of Stammler triangle
X(6) = eigencenter of outer Grebe triangle
X(6) = eigencenter of inner Grebe triangle
X(6) = eigencenter of submedial triangle
X(6) = perspector of unary cofactor triangles of every pair of homothetic triangles
X(6) = perspector of ABC and unary cofactor triangle of any triangle homothetic to ABC
X(6) = perspector of Stammler triangle and unary cofactor triangle of circumtangential triangle
X(6) = perspector of Stammler triangle and unary cofactor triangle of circumnormal triangle
X(6) = perspector of submedial triangle and unary cofactor triangle of orthic triangle
X(6) = perspector of unary cofactor triangles of extraversion triangles of X(7) and X(9)
X(6) = X(3)-of-reflection-triangle-of-X(2)
X(6) = center of the orthic inconic
X(6) = orthic isogonal conjugate of X(25)
X(6) = center of bicevian conic of X(371) and X(372)
X(6) = center of bicevian conic of X(6) and X(25)
X(6) = perspector of ABC and mid-triangle of Mandart-incircle and Mandart-excircles triangles
X(6) = X(381)-of-anti-Artzt-triangle
X(6) = homothetic center of medial triangle and cross-triangle of ABC and inner Grebe triangle
X(6) = homothetic center of medial triangle and cross-triangle of ABC and outer Grebe triangle
X(6) = 4th-anti-Brocard-to-anti-Artzt similarity image of X(3)
X(6) = perspector of pedal and anticevian triangles of X(3)
X(6) = X(9)-of-orthic-triangle if ABC is acute
X(6) = X(7)-of-tangential-triangle if ABC is acute
X(6) = X(53)-of-excentral-triangle
X(6) = Thomson-isogonal conjugate of X(376)
X(6) = perspector of ABC and mid-triangle of 1st and 2nd anti-Conway triangles
X(6) = X(193)-of-3rd-tri-squares-central-triangle
X(6) = X(193)-of-4th-tri-squares-central-triangle
X(6) = X(6)-of-circumsymmedial-triangle
X(6) = X(6)-of-inner-Grebe-triangle
X(6) = X(6)-of-outer-Grebe-triangle
X(6) = X(157)-of-intouch-triangle
X(6) = perspector, wrt Schroeter triangle, of polar circle
X(6) = center of the perspeconic of these triangles: ABC and Ehrmann vertex
X(6) = barycentric square of X(1)
X(6) = pole, wrt circumcircle, of Lemoine axis
X(6) = pole wrt polar circle of trilinear polar of X(264) (line X(297)X(525))
X(6) = polar conjugate of X(264)
X(6) = X(i)-isoconjugate of X(j) for these {i,j}: {1,2}, {6,75}, {31,76}, {91,1993}, {110, 1577}, {338,1101}, {1994,2962}
X(6) = inverse-in-2nd-Brocard-circle of X(39)
X(6) = inverse-in-Steiner-inellipse of X(230)
X(6) = inverse-in-Steiner-circumellipse of X(385)
X(6) = inverse-in-Kiepert-hyperbola of X(381)
X(6) = inverse-in-circumconic-centered-at-X(9) of X(238)
X(6) = perspector of medial triangle and half-altitude triangle
X(6) = intersection of tangents to Kiepert hyperbola at X(2) and X(4)
X(6) = antigonal conjugate of X(67)
X(6) = vertex conjugate of foci of Steiner inellipse
X(6) = X(99)-of-1st-Brocard-triangle
X(6) = X(1379)-of-2nd-Brocard-triangle
X(6) = X(6)-of-4th-Brocard-triangle
X(6) = X(6)-of-orthocentroidal-triangle
X(6) = reflection of X(2453) in the Euler line
X(6) = similitude center of ABC and orthocentroidal triangle
X(6) = similitude center of 4th Brocard and circumsymmedial triangles
X(6) = tangential isogonal conjugate of X(22)
X(6) = tangential isotomic conjugate of X(1498)
X(6) = barycentric product of (nonreal) circumcircle intercepts of the line at infinity
X(6) = eigencenter of anti-orthocentroidal triangle
X(6) = perspector of Aquarius conic
X(6) = trilinear pole wrt tangential triangle of Lemoine axis
X(6) = trilinear pole wrt symmedial triangle of Lemoine axis
X(6) = trilinear pole wrt circumsymmedial triangle of Lemoine axis
X(6) = crosspoint of X(2) and X(194) wrt both the excentral and anticomplementary triangles
X(6) = pedal antipodal perspector of X(5004) and of X(5005)
X(6) = vertex conjugate of Jerabek hyperbola intercepts of Lemoine axis
X(6) = hyperbola {{A,B,C,:ref:X(2) <X(2)>,:ref:X(6) <X(6)>}} antipode of X(694)
X(6) = perspector of orthic triangle and tangential triangle, wrt orthic triangle, of the circumconic of the orthic triangle centered at X(4) (the bicevian conic of X(4) and X(459))
X(6) = perspector of excentral triangle and extraversion triangle of X(9)
X(6) = excentral-to-ABC functional image of X(9)
X(6) = orthic-to-ABC functional image of X(53)
X(6) = intouch-to-ABC functional image of X(7)
X(6) = 1st-Brocard-isogonal conjugate of X(804)
X(6) = center of the MacBeath circumconic
X(6) = center of the cosine circle (the 2nd Lemoine circle)
X(6) = one of the foci of the Lemoine inellipse (the other being X(2))
X(6) = antipode of X(32113) in Walsmith rectangular hyperbola
X(6) = orthocenter of X(74)X(110)X(3569)
X(6) = orthocenter of X(113)X(125)X(3569)
X(6) = QA-P23 (Inscribed Square Axes Crosspoint) of quadrangle ABC:ref:X(2) <X(2)>; see http://www.chrisvantienhoven.nl/quadrangle-objects/15-mathematics/quadrangle-objects/artikelen-qa/51-qa-p23.html