X401–X484

Euler Line Substitutions & Hirst Inverses

Centers lying on the Euler line generated by Xk-substitutions and Hirst inversions, followed by the Tixier point, Ajima-Malfatti radical center, and Evans perspectors.
< A B C >: Reference Triangle △ABC [ A B ]: Sideline AB (Base c = 1) ( A R_b ): Euclid I.22 Guide Circle (A) ( B R_a ): Euclid I.22 Guide Circle (B) [ B C ]: Sideline BC (Side a) [ C A ]: Sideline CA (Side b) circumcircle: Circumcircle of △ABC incircle: Incircle of △ABC nine_point_circle: Nine-Point Circle (Euler / Feuerbach Circle) { 2, 3 }: Euler Line A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) R_b = Radius Point R_b (0.8571, 0.0000) R_a = Radius Point R_a (0.2857, 0.0000) C = Vertex C (0.6122, 0.5999) X_3 = CIRCUMCENTER (0.5000, 0.1021) X_1 = INCENTER (0.5714, 0.2333) X_5 = NINE-POINT CENTER (0.5561, 0.2489) X_2 = CENTROID (0.5374, 0.2000) X_402 = ZEEMAN-GOSSARD PERSPECTOR (0.5786, 0.3077) X_403 = OF THE ORTHIC TRIANGLE (0.7040, 0.6358) X_405 = EULER LINE INTERCEPT OF LINE X(1)X(6) (0.5325, 0.1870) X_408 = EULER LINE INTERCEPT OF LINE X(73)X(228) (0.4882, 0.0713) X_411 = EULER X(21)-2nd-SUBSTITUTION POINT (0.4414, -0.0514) X_412 = EULER X(29)-2nd-SUBSTITUTION POINT (0.6482, 0.4899) X_415 = -HIRST INVERSE OF X(29) (0.7758, 0.8236) X_423 = -HIRST INVERSE OF X(27) (0.7431, 0.7380) X_424 = -HIRST INVERSE OF X(451) (0.8111, 0.9162) X_425 = -HIRST INVERSE OF X(21) (0.8561, 1.0339) X_430 = EULER X(81)-5th-SUBSTITUTION POINT (0.6091, 0.3875) X_441 = COMPLEMENT OF X(297) (0.3841, -0.2011) X_450 = -HIRST INVERSE OF X(4) (0.8978, 1.1430) X_476 = TIXIER POINT (0.9824, 0.2685) X_477 = TIXIER ANTIPODE (0.0176, -0.0644) X_478 = CENTER OF YIU CONIC (0.5030, -0.3520) A B C X3 X1 X5 X2 X402 X403 X405 X408 X411 X412 X415 X423 X424 X425 X430 X441 X450 X476 X477 X478 (a, b, c) = (5, 6, 7) | AB = 1

X401–X484 (Euler Line Substitutions & Hirst Inverses) — 16 representative centers (from 84 in collection) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

X_402 — ZEEMAN-GOSSARD PERSPECTOR

X402ZEEMAN-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).
X_413 — EULER X_21-3rd-SUBSTITUTION POINT

X413EULER X21-3rd-SUBSTITUTION POINT

As a point on the Euler line, X413 has Shinagawa coefficients \((\$aS_{A}\$[(\$aS_{A}\$)^{2} - \$bcS_{B}S_{C}\$] - 2abcFS^{2}, -(\$aS_{A}\$ - abc)[(\$aS_{A}\$)^{2} - \$bcS_{B}S_{C}\$] - 2[\$bcS_{B}S_{C}\$ + \$aS_{A}\$F - abcF]S^{2})\).

X_414 — EULER X_29-3rd-SUBSTITUTION POINT

X414EULER X29-3rd-SUBSTITUTION POINT

As a point on the Euler line, X414 has Shinagawa coefficients \(([(2E-F)F-S^{2}]FS^{4} -\$abS_{A}S_{B}\$F^{2}S^{2} +\$ab\$F^{2}S^{4}, -[(3E-F)F-S^{2}]FS^{4} -\$ab(S_{A}S_{B})^{3}\$ +2\$abS_{C}^{3}\$FS^{2} -4\$abS_{A}S_{B}\$F^{2}S^{2} -2\$abS_{C}\$[(E+F)^{2}-2S^{2}]FS^{2} -\$ab\$F^{2}S^{4})\).

X_428 — EULER X_38-5th-SUBSTITUTION POINT

X428EULER X38-5th-SUBSTITUTION POINT

Let H = X4 and let HaHbHc be the pedal triangle of H in the plane of a triangle ABC. Let

  • Bc = reflection of Hb in HHc, and define Ca and Ab cyclically;

  • Cb = reflection of Hc in HHb, and define Ac and Ba cyclically;

  • Ha = orthogonal projecton of A on BcCb, and define Hb and Hc cyclically.

Then X428 = centroid of triangle HaHbHc. See Antreas Hatzipolakis and Peter Moses, Hyacinthos 27169

X_435 — EULER X_399-6th-SUBSTITUTION POINT

X435EULER X399-6th-SUBSTITUTION POINT

As a point on the Euler line, X435 has Shinagawa coefficients \((9E^{2}F + 144EF^{2} - 64FS^{2}, -9E^{3} + 89E^{2}F - 128EF^{2} - 64F^{3} + 32(E - 2F)S^{2})\)

X_437 — EULER X_214-7th-SUBSTITUTION POINT

X437EULER X214-7th-SUBSTITUTION POINT

As a point on the Euler line, X437 has Shinagawa coefficients \((2(E+F)^{2}F+10FS^{2}+7\$bcS_{A}\$F-5\$ab\$(E+F)F, (5E-16F)S^{2}-\$abS_{C}^{2}\$ +41\$bcS_{B}S_{C}\$+\$ab\$[(E+F)^{2}-19S^{2}])\).

X_453 — POINT ALSHAIN

X453POINT ALSHAIN

As a point on the Euler line, X453 has Shinagawa coefficients \((2\$aS_{B}S_{C}\$+\$aS_{A}\$(E+2F)-2\$a\$S^{2}-2abcF, -\$aS_{A}\$E+2\$a\$S^{2}+abcE)\).

X_476 — TIXIER POINT

X476TIXIER POINT

The reflection of X110 in the Euler line; X476 is on the circumcircle. (Michel Tixier, 5/9/98). Also, X476 is the center of the polar conic of X30 with respect to the Neuberg cubic; this conic is a rectangular hyperbola passing through the incenter, the excenters, and X30. (Peter Yff, 5/23/99)
X_478 — CENTER OF YIU CONIC

X478CENTER OF YIU CONIC

Center of the Yiu conic, which passes through the points outside the circumcircle at which the excircles of ABC are tangent to the sidelines of ABC. See Paul Yiu’s
X_479 — X_269-CROSS CONJUGATE OF X_279

X479-CROSS CONJUGATE OF X279

Let A' be the point in which the incircle is tangent to a circle that passes through vertices B and C, and define B' and C' cyclically. The lines AA', BB', CC' concur in X479. For an excircle version, see X5423.
X_481 — 1st EPPSTEIN POINT

X4811st EPPSTEIN POINT

Let S be the inner Soddy circle and \(S_{A}\), \(S_{B}\), \(S_{C}\) the Soddy circles tangent to S. Let Ia = S∩\(S_{A}\), Ea = \(S_{B}\)∩\(S_{C}\), and determine Ib, Ic, Eb, Ec cyclically. Then X481 is the point of concurrence of lines IA-to-EA, IB-to-EB, IC-to-EC.
X_482 — 2nd EPPSTEIN POINT

X4822nd EPPSTEIN POINT

Let S' be the outer Soddy circle and \(S_{A}\), \(S_{B}\), \(S_{C}\) the Soddy circles tangent to S. Let JA = S'∩\(S_{A}\), EA = \(S_{B}\)∩\(S_{C}\), and determine JB, JC, EB, EC cyclically. Then X482 is the point of concurrence of lines JA-to-EA, JB-to-EB, JC-to-EC.
X_484 — 1st EVANS PERSPECTOR

X4841st EVANS PERSPECTOR

X484 is the perspector of the excentral triangle and the triangle RefaRefbRefc, where Refa is the reflection of vertex A in sideline BC and Refb, Refc are determined cyclically. (Lawrence Evans, 10/22/98)