X612

INTERSECTION OF LINES X1X_2 AND X9X_31

X612 is the homothetic center of the incentral triangle and the Ayme triangle; see X3610.
< 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) { 1, 2 }: Nagel Line { 9, 31 }: Central Line L(9, 31) { 19, 25 }: Central Line L(19, 25) { 12, 34 }: Central Line L(12, 34) { 22, 35 }: Central Line L(22, 35) { 38, 57 }: Central Line L(38, 57) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_1 = INCENTER (0.5714, 0.2333) X_2 = CENTROID (0.5374, 0.2000) X_9 = MITTENPUNKT (0.5110, 0.1615) X_31 = 2nd POWER POINT (0.6228, 0.3008) X_19 = CLAWSON POINT (0.6246, 0.4244) X_25 = HOMOTHETIC CENTER OF ORTHIC AND TANGENTIAL TRIANGLES (0.6329, 0.4497) X_12 = {X(1),X(5)}-HARMONIC CONJUGATE OF X(11) (0.5641, 0.2407) X_34 = -BETH CONJUGATE OF X(4) (0.6332, 0.4792) X_22 = EXETER POINT (0.4143, -0.1221) X_35 = {X(1),X(3)}-HARMONIC CONJUGATE OF X(36) (0.5373, 0.1706) X_38 = CROSSPOINT OF X(1) AND X(75) (0.5443, 0.1976) X_57 = ISOGONAL CONJUGATE OF X(9) (0.6138, 0.3110) X_612 = INTERSECTION OF LINES X(1)X(2) AND X(9)X(31) (0.5582, 0.2203) A B C X1 X2 X9 X31 X19 X25 X12 X34 X22 X35 X38 X57 X612 (a, b, c) = (5, 6, 7) | AB = 1

X612 (INTERSECTION OF LINES X1X_2 AND X9X_31) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \(a^{2} + b^{2} + c^{2} + 2bc\) (M. Iliev, 5/13/2007)
Search Value (6, 9, 13): \(2.013984607013\)
▸ Alternate Coordinates3 forms

  • Trilinears: \(bc + S_{\omega } : ca + S_{\omega } : ab + S_{\omega }\) (C. Lozada, 9/07/2013)

  • Trilinears: \((1 + \cos A)(1 + \cos B \cos C) : :\)

  • Trilinears: \((1 + \sec A)(1 + \sec B \sec C) : :\)

▸ Incident Central Lines14 lines

{1, 2}, {6, 210}, {9, 31}, {12, 34}, {19, 25}, {21, 989}, {22, 35}, {38, 57}, {63, 171}, {165, 990}, {394, 611}, {404, 988}, {495, 1060}, {518, 940}

▸ Functional Relations3 relations