X1109

TRILINEAR SQUARE OF X523

X1109 = crosspoint of X12 and X523
< 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) { 31, 92 }: Central Line L(31, 92) { 91, 255 }: Central Line L(91, 255) { 11, 523 }: Central Line L(11, 523) { 1, 564 }: Central Line L(1, 564) { 75, 799 }: Central Line L(75, 799) { 12, 1091 }: Central Line L(12, 1091) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_1101 = TRILINEAR SQUARE OF X(110) (0.5344, 0.4306) X_31 = 2nd POWER POINT (0.6228, 0.3008) X_92 = CEVAPOINT OF INCENTER AND CLAWSON POINT (0.5949, 0.3639) X_91 = ISOGONAL CONJUGATE OF X(47) (-0.0865, -0.0205) X_255 = ISOGONAL CONJUGATE OF X(158) (0.4217, 0.0365) X_11 = FEUERBACH POINT (0.7347, 0.0667) X_523 = ISOGONAL CONJUGATE OF X(110) (1.0856, -0.0096) X_1 = INCENTER (0.5714, 0.2333) X_564 = INTERSECTION OF LINES X(1)X(1048) AND X(47,91) (0.4993, 0.3100) X_75 = ISOTOMIC CONJUGATE OF INCENTER (0.4988, 0.1682) X_799 = ISOGONAL CONJUGATE OF X(798) (0.0470, 0.3636) X_12 = {X(1),X(5)}-HARMONIC CONJUGATE OF X(11) (0.5641, 0.2407) X_1091 = TRILINEAR SQUARE OF X(12) (0.5571, 0.2482) X_1109 = TRILINEAR SQUARE OF X(523) (0.7246, 0.0705) A B C X1101 X31 X92 X91 X255 X11 X523 X1 X564 X75 X799 X12 X1091 X1109 (a, b, c) = (5, 6, 7) | AB = 1

X1109 (TRILINEAR SQUARE OF X523) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \(\sin^{2}(B - C) : :\)
Search Value (6, 9, 13): \(2.983960439511\)
▸ Alternate Coordinates1 form

  • Trilinears: \(1 - \cos(2B - 2C) : :\)

▸ Incident Central Lines6 lines

{1, 564}, {11, 523}, {12, 1091}, {31, 92}, {75, 799}, {91, 255}

▸ Functional Relations14 relations

Notes

X1109: Let EuaEubEuc be the Feuerbach triangle. Let La be the trilinear polar of Eua, and define Lb, Lc cyclically. Let A″ = Lb∩Lc, B″ = Lc∩La, C″ = La∩Lb. The lines AA″, BB″, CC″ concur in X1109. (Randy Hutson, December 26, 2015)

Let EuaEubEuc be the Feuerbach triangle. Let Ma be the tangent to conic {{A, B, C,Eub,Euc}} at A, and define Mb and Mc cyclically. Let A* = Mb∩Mc, B* = Mc∩Ma, C* = Ma∩Mb. The lines AA*, BB*, CC* concur in X1109; see also X523. (Randy Hutson, December 26, 2015)

Let EuaEubEuc be the Feuerbach triangle. Let Ab = BC∩EucEua, and define Bc and Ca cyclically. Let Ac = BC∩EuaEub, and define Ba and Cb cyclically. The points Ab, Ac, Bc, Ba, Ca, Cb lie on an ellipse, denoted by E. Let A″ be the intersection of the tangents to E at Ba and Ca, and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X1109. (Randy Hutson, December 26, 2015)

Let EuaEubEuc be the Feuerbach triangle. Let A″ be the trilinear product Eub*Euc, and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X1109. (Randy Hutson, December 26, 2015)

Let La be the A-extraversion of line X2610X4024 (the trilinear polar of X12), and define Lb and Lc cyclically. Let A' = Lb ∩ Lc, and define B' and C' cyclically. The lines AA', BB', CC' concur in X1109. (Randy Hutson, January 29, 2018)

Let F be the Feuerbach point, X11, and FaFbFc be the Feuerbach triangle (the extraversion triangle of X11). Let A' be the trilinear product F*Fa, and define B' and C' cyclically. The lines AA', BB', CC' concur in X1109. (Randy Hutson, January 29, 2018)