X3086

INTERSECTION OF LINES X1X_2 AND X4X_11

For a construction and relationships to other points, see the preamble just before X37994.
< 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 { 4, 11 }: Central Line L(4, 11) { 20, 36 }: Central Line L(20, 36) { 7, 90 }: Central Line L(7, 90) { 5, 388 }: Central Line L(5, 388) { 35, 390 }: Central Line L(35, 390) 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_4 = ORTHOCENTER (0.6122, 0.3958) X_11 = FEUERBACH POINT (0.7347, 0.0667) X_20 = DE LONGCHAMPS POINT (0.3878, -0.1916) X_36 = INVERSE-IN-CIRCUMCIRCLE OF INCENTER (1.3333, 1.6330) X_7 = GERGONNE POINT (0.5903, 0.2769) X_90 = -CROSS CONJUGATE OF X(1) (0.1022, 0.0201) X_5 = NINE-POINT CENTER (0.5561, 0.2489) X_388 = ∩X(7)X(8) (0.5790, 0.2635) X_35 = {X(1),X(3)}-HARMONIC CONJUGATE OF X(36) (0.5373, 0.1706) X_390 = REFLECTION OF GERGONNE POINT IN INCENTER (0.5526, 0.1897) X_3086 = INTERSECTION OF LINES X(1)X(2) AND X(4)X(11) (0.6456, 0.3060) A B C X1 X2 X4 X11 X20 X36 X7 X90 X5 X388 X35 X390 X3086 (a, b, c) = (5, 6, 7) | AB = 1

X3086 (INTERSECTION OF LINES X1X_2 AND X4X_11) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \(1 - \sin B \sin C : :\)
Barycentrics: \(a (4 R^{2} - b c) : b (4 R^{2} - c a) : c (4 R^{2} - a b)\) (Vijay Krishna, April 15, 2020)
Search Value (6, 9, 13): \(1.202484985804\)
▸ Incident Central Lines28 lines

{1, 2}, {3, 496}, {4, 11}, {5, 388}, {7, 90}, {12, 1056}, {20, 36}, {31, 3075}, {33, 3088}, {34, 3089}, {35, 390}, {55, 631}, {57, 946}, {69, 613}, {171, 1497}, {238, 255}, {269, 1256}, {406, 1104}, {443, 2886}, {495, 1656}, {602, 1936}, {603, 3073}, {748, 1496}, {944, 1319}, {1124, 3068}, {1335, 3069}, {1588, 2067}, {2242, 2548}

▸ Functional Relations3 relations

Notes

Let Ab =\(0 : 2R : c\), Ac =\(0 : b : 2R\), Bc =\(a : 0 : 2R\), Ba =\(2R : 0 : c\), Ca =\(2R : b : 0\), Cb =\(a : 2R : 0\) be the six points in the construction of the Paasche conic at X37861 and X37881. Let Ka be the point of intersection of the lines tangent to the Paasche conic at Ba and Ca, and define Kb and Kc cyclically. The triangle KaKbKc is perspective to ABC, and the perspector is X3086; see also X37884 and X3086 (X(3086).pdf) . (Vijay Krishna, April 15, 2020)