X402

ZEEMAN-GOSSARD PERSPECTOR

In A History of Mathematics, Florian Cajori writes, “H. C. Gossard of the University of Oklahoma showed in 1916 that the three Euler lines of the triangles formed by the Euler line and the sides, taken by twos, of a given triangle, form a triangle … perspective with the given triangle and having the same Euler line.” Let ABC be the given triangle and A'B'C' the Gossard triangle - that is, the triangle perspective with the given triangle and having the same Euler line. The lines AA', BB', CC' concur in X402, named the Gosssard perspector by John Conway (1998).
< A B C >: Reference Triangle △ABC [ A B ]: Sideline AB (Base c = 1) [ B C ]: Sideline BC (Side a) [ C A ]: Sideline CA (Side b) { 2, 3 }: Euler Line A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_2 = CENTROID (0.5374, 0.2000) X_3 = CIRCUMCENTER (0.5000, 0.1021) X_4 = ORTHOCENTER (0.6122, 0.3958) X_402 = ZEEMAN-GOSSARD PERSPECTOR (0.5786, 0.3077) A B C X2 X3 X4 X402 (a, b, c) = (5, 6, 7) | AB = 1

X402 (ZEEMAN-GOSSARD PERSPECTOR) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Barycentrics: \((2 a^{4} - a^{2} b^{2} - b^{4} - a^{2} c^{2} + 2 b^{2} c^{2} - c^{4}) \cdot (a^{8} - a^{6} b^{2} - 2 a^{4} b^{4} + 3 a^{2} b^{6} - b^{8} - a^{6} c^{2} + 5 a^{4} b^{2} c^{2} - 3 a^{2} b^{4} c^{2} - b^{6} c^{2} - 2 a^{4} c^{4} - 3 a^{2} b^{2} c^{4} + 4 b^{4} c^{4} + 3 a^{2} c^{6} - b^{2} c^{6} - c^{8}) : :\) (Paul Yiu, February 20, 1999)
Search Value (6, 9, 13): \(2.073784975589\)
▸ Linear Combinations1 combination

  • X402 = X1 - \(3\) X11831, X1 + \(3\) X11852, \(5\) X2 + X3081, \(3\) X2 + X4240, \(5\) X2 - X11050, \(3\) X2 - \(5\) X15183, \(2\) X2 + X34582, X3 + \(3\) X11911, X3 - \(3\) X26451, \(3\) X3 - X35241, X4 + \(3\) X11845, X4 - \(3\) X11897, \(2\) X5 + X15774, X8 - \(3\) X16210, X8 + \(3\) X16212, X20 - \(3\) X16190, X145 - \(3\) X16211, \(3\) X381 - X18507, \(3\) X381 + X18508, X1650 + \(3\) X1651, \(5\) X1650 + \(3\) X3081, \(2\) X1650 - \(3\) X11049, \(5\) X1650 - \(3\) X11050, X1650 - \(5\) X15183, \(2\) X1650 + \(3\) X34582, \(5\) X1651 - X3081, \(3\) X1651 - X4240, \(2\) X1651 + X11049, \(5\) X1651 + X11050, \(3\) X1651 + \(5\) X15183, \(3\) X1651 + \(2\) X15184, \(3\) X3081 - \(5\) X4240, \(2\) X3081 + \(5\) X11049, \(3\) X3081 + \(25\) X15183, \(3\) X3081 + \(10\) X15184, \(2\) X3081 - \(5\) X34582, \(2\) X4240 + \(3\) X11049, \(5\) X4240 + \(3\) X11050, X4240 + \(5\) X15183, X4240 + \(2\) X15184, \(2\) X4240 - \(3\) X34582, \(3\) X5055 + X20128, \(5\) X11049 - \(2\) X11050, \(3\) X11049 - \(10\) X15183, \(3\) X11049 - \(4\) X15184, \(3\) X11050 - \(25\) X15183, \(3\) X11050 - \(10\) X15184, \(2\) X11050 + \(5\) X34582, X11251 - \(3\) X11911, X11251 + \(3\) X26451, \(3\) X11251 + X35241, \(3\) X11831 + X12438, \(3\) X11845 - X12113, \(3\) X11852 - X12438, \(3\) X11897 + X12113, \(9\) X11911 + X35241, X12626 + \(3\) X16210, X12626 - \(3\) X16212, \(5\) X15183 - \(2\) X15184, \(10\) X15183 + \(3\) X34582, \(4\) X15184 + \(3\) X34582, \(9\) X26451 - X35241 (Peter Moses, November 26,2020)

▸ Incident Central Lines62 lines

{1, 11831}, {2, 3}, {6, 11901}, {8, 12626}, {10, 11900}, {11, 11903}, {12, 11904}, {13, 12793}, {14, 12792}, {17, 22897}, {18, 22852}, {32, 11839}, {40, 12696}, {54, 12797}, {55, 11848}, {56, 18958}, {64, 12791}, {67, 32279}, {68, 12418}, {74, 12369}, {76, 12794}, {79, 16129}, {80, 12729}, {83, 12795}, {84, 12668}, {98, 12181}, {99, 13179}, {100, 13268}, {104, 12752}, {110, 13212}, {112, 13281}, {113, 38605}, {145, 16211}, {190, 24830}, {262, 22698}, {265, 12790}, {371, 35791}, {372, 35790}, {485, 12800}, {486, 12799}, {493, 11907}, {494, 11908}, {671, 12347}, {1297, 12796}, {1327, 13689}, {1328, 13809}, {3068, 13894}, {3069, 13948}, {5597, 11863}, {5598, 11864}, {5877, 26958}, {5972, 6130}, {6145, 32372}, {6776, 39886}, {7160, 12789}, {7740, 18279}, {10266, 12798}, {11657, 16186}, {13494, 13495}, {22466, 22943}, {25641, 38625}, {34297, 38794}

▸ Functional Relations13 relations

  • X402 = midpoint of Xi and Xj for these {i,j}: {1, 12438}, {2, 1651}, {3, 11251}, {4, 12113}, {5, 32162}, {6, 12583}, {8, 12626}, {13, 12793}, {14, 12792}, {17, 22897}, {18, 22852}, {40, 12696}, {54, 12797}, {64, 12791}, {67, 32279}, {68, 12418}, {74, 12369}, {76, 12794}, {79, 16129}, {80, 12729}, {83, 12795}, {84, 12668}, {98, 12181}, {99, 13179}, {100, 13268}, {104, 12752}, {110, 13212}, {112, 13281}, {190, 24830}, {262, 22698}, {265, 12790}, {402, 402}, {485, 12800}, {486, 12799}, {671, 12347}, {1297, 12796}, {1327, 13689}, {1328, 13809}, {1650, 4240}, {3081, 11050}, {6145, 32372}, {6776, 39886}, {7160, 12789}, {7740, 18279}, {10266, 12798}, {11049, 34582}, {11831, 11852}, {11832, 11853}, {11839, 11885}, {11845, 11897}, {11848, 11909}, {11863, 26383}, {11864, 26407}, {11901, 19017}, {11902, 19018}, {11903, 11906}, {11904, 11905}, {11907, 26447}, {11908, 26448}, {11911, 26451}, {11912, 26452}, {11913, 26453}, {11914, 11915}, {13494, 13495}, {13894, 26449}, {13948, 26450}, {16210, 16212}, {18507, 18508}, {18958, 22755}, {22466, 22943}

  • X402 = reflection of Xi in Xj for these {i,j}: {11049, 2}, {34582, 1651}, {1650, 15184}, {15774, 32162}

  • X402 = complement of X1650

  • X402 = anticomplement of X15184

  • X402 = isotomic conjugate of the anticomplement of X32750

  • X402 = Xi-complementary conjugate of Xj for these (i,j): {162, 16177}, {1304, 34846}, {15459, 21253}, {24000, 113}, {32640, 16595}, {32676, 39008}, {32695, 8287}, {32715, 16573}, {36034, 122}, {36117, 37985}, {36131, 15526}

  • X402 = Xi-Ceva conjugate of Xj for these (i,j): {5972, 31945}, {9033, 30}, {13494, 523}

  • X402 = X32750-cross conjugate of X2

  • X402 = Xi-isoconjugate of Xj for these (i,j): {74, 9390}, {1304, 9392}, {2159, 15351}

  • X402 = Xi-reciprocal conjugate of Xj for these (i,j): {30, 15351}, {1494, 39352}, {2173, 9390}, {2349, 2629}, {2631, 9392}, {15184, 32750}, {16077, 39062}, {34767, 38240}

  • X402 = barycentric product Xi*Xj for these {i,j}: {30, 39352}, {2629, 14206}, {4240, 38240}, {9033, 39062}

  • X402 = barycentric quotient Xi/Xj for these {i,j}: {30, 15351}, {2173, 9390}, {2629, 2349}, {2631, 9392}, {32750, 15184}, {38240, 34767}, {39062, 16077}, {39352, 1494}

  • X402 = {Xi,Xj}-harmonic conjugate of Xk for these (i,j,k): {1, 11852, 12438}, {1, 26452, 11915}, {1, 26453, 11914}, {2, 1650, 15184}, {2, 4240, 1650}, {3, 11911, 11251}, {4, 11845, 12113}, {8, 16212, 12626}, {381, 18508, 18507}, {1650, 1651, 4240}, {1650, 15183, 2}, {1650, 15184, 11049}, {1651, 15183, 1650}, {4240, 15183, 15184}, {11251, 26451, 3}, {11831, 12438, 1}, {11897, 12113, 4}, {11901, 11902, 12583}, {11912, 11913, 1}, {11912, 12438, 11914}, {11913, 12438, 11915}, {12626, 16210, 8}, {19017, 19018, 6}, {26452, 26453, 12438}

Notes

As a point on the Euler line, X402 has Shinagawa coefficients \((5EF - 13F^{2} - S^{2}, 3EF + 3F^{2} - S^{2})\).

Actually, X402 dates back to an article by Christopher Zeeman in Wiskundige Opgaven 8 (1899-1902) 305. For details, see Paul Yiu’s Hyacinthos message #7536 and others with Gossard in the subject line. (In ETC, the change of name from Gossard Perspector to Zeeman-Gossard Perspector was made on Oct. 15, 2003.) Further details are given by Wilson Stothers in Hyacinthos #8383, Oct. 21, 2003.

(From Angel Montesdeoca, November 25, 2020) Let P be a variable point on the Euler line and Q its isotomic conjugate. Let TP and TQ be the cevian triangles of P and Q, respectively. Let F be the finite fixed point of the affine transformation that carries TP onto TQ. The envelope of the line PF is a hyperbola (H) with center X402, an asymptote of the Euler line. A barycentric equation for (H) follows:

cyclic sum [(-b^4+c^4+a^2 (b^2-c^2))^4 (-2 a^4+(b^2-c^2)^2+a^2 (b^2+c^2))^2x^2-2 (a^2-b^2)^2 (a^2-c^2)^2 (a^4-(b^2-c^2)^2)^2 (a^8-a^6 (b^2+c^2)+5 a^2 (b^2-c^2)^2 (b^2+c^2)+a^4 (-3 b^4+7 b^2 c^2-3 c^4)-(b^2-c^2)^2 (2 b^4+5 b^2 c^2+2 c^4))y z ] = 0.