X393

X25-CROSS CONJUGATE OF X4

X393 = isogonal conjugate of X394
< 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) { 4, 6 }: Van Aubel Line { 33, 42 }: Central Line L(33, 42) { 37, 158 }: Central Line L(37, 158) { 2, 216 }: Central Line L(2, 216) { 107, 111 }: Central Line L(107, 111) { 19, 208 }: Central Line L(19, 208) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_25 = HOMOTHETIC CENTER OF ORTHIC AND TANGENTIAL TRIANGLES (0.6329, 0.4497) X_4 = ORTHOCENTER (0.6122, 0.3958) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6000, 0.2672) X_33 = PERSPECTOR OF THE ORTHIC AND INTANGENTS TRIANGLES (0.6019, 0.3546) X_42 = CROSSPOINT OF INCENTER AND SYMMEDIAN POINT (0.5880, 0.2495) X_37 = CROSSPOINT OF INCENTER AND CENTROID (0.5567, 0.2158) X_158 = -CROSS CONJUGATE OF X(92) (0.6215, 0.5117) X_2 = CENTROID (0.5374, 0.2000) X_216 = -CEVA CONJUGATE OF X(51) (0.5201, 0.1353) X_107 = Ψ(SYMMEDIAN POINT, ORTHOCENTER) (0.5733, 0.6071) X_111 = PARRY POINT (0.9885, -0.0457) X_19 = CLAWSON POINT (0.6246, 0.4244) X_208 = -CEVA CONJUGATE OF X(34) (0.6242, -0.2728) X_393 = -CROSS CONJUGATE OF X(4) (0.6247, 0.5263) A B C X25 X4 X6 X33 X42 X37 X158 X2 X216 X107 X111 X19 X208 X393 (a, b, c) = (5, 6, 7) | AB = 1

X393 (X25-CROSS CONJUGATE OF X4) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \(bc \tan^{2} A : ca \tan^{2} B : ab \tan^{2} C\)
Barycentrics: \(\tan^{2} A : \tan^{2} B : \tan^{2} C\)
Search Value (6, 9, 13): \(0.377154310128\)
▸ Alternate Coordinates6 forms

  • Trilinears: \(\sec A - \csc B \csc C : \sec B - \csc C \csc A : \sec C - \csc A \csc B\)

  • Trilinears: \(\sec A \tan A : :\)

  • Trilinears: \(1/(\sin A - \csc A) : :\)

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

  • Barycentrics: \([a^{4} - (b^{2} - c^{2})^{2}]^{2} : :\)

  • Barycentrics: \(\tan A - \cot B - \cot C : :\)

▸ Incident Central Lines19 lines

{1, 836}, {2, 216}, {4, 6}, {19, 208}, {20, 577}, {24, 254}, {25, 1033}, {27, 967}, {33, 42}, {37, 158}, {69, 297}, {107, 111}, {193, 317}, {230, 459}, {278, 1108}, {342, 948}, {394, 837}, {800, 1093}, {3146, 3284}

▸ Functional Relations14 relations

Notes

Let Ha be the foot of the A-altitude of ABC. Let Pa be the foot of the altitude from Ha to AB, and Qa the foot of the altitude from Ha to CA. Define Hb, Hc, Pb, Pc, Qb, Qc cyclically. Let A' be the trilinear pole of line PaQa, and define B' and C' cyclically. The lines AA', BB', CC' concur in X393. Lines PaQa, PbQb, PcQc are the antiparallels in the construction of the Taylor circle. (Randy Hutson, January 29, 2018)

Let A'B'C' and A″B″C″ be the Euler and anti-Euler triangles, resp. Let A* be the barycentric product A'*A″, and define B* and C* cyclically. The lines AA*, BB*, CC* concur in X393. (Randy Hutson, January 29, 2018)

Let HaHbHc be the orthic triangle. Let A″ be the cevapoint of the (real or imaginary) circumcircle intercepts of line HbHc. Define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X393. (Randy Hutson, July 31 2018)

X393 is the Brianchon point (perspector) of the inellipse that is the barycentric square of the orthic axis. The center of this inellipse is X3767. (Randy Hutson, October 15, 2018)