X3085

INTERSECTION OF LINES X1X_2 AND X4X_12

X3085 = {X1,X2}-harmonic conjugate of X3086
< 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, 12 }: Central Line L(4, 12) { 7, 46 }: Central Line L(7, 46) { 20, 35 }: Central Line L(20, 35) { 37, 158 }: Central Line L(37, 158) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_3086 = INTERSECTION OF LINES X(1)X(2) AND X(4)X(11) (0.6456, 0.3060) X_1 = INCENTER (0.5714, 0.2333) X_2 = CENTROID (0.5374, 0.2000) X_4 = ORTHOCENTER (0.6122, 0.3958) X_12 = {X(1),X(5)}-HARMONIC CONJUGATE OF X(11) (0.5641, 0.2407) X_7 = GERGONNE POINT (0.5903, 0.2769) X_46 = -CEVA CONJUGATE OF X(1) (0.6917, 0.4543) X_20 = DE LONGCHAMPS POINT (0.3878, -0.1916) X_35 = {X(1),X(3)}-HARMONIC CONJUGATE OF X(36) (0.5373, 0.1706) X_37 = CROSSPOINT OF INCENTER AND CENTROID (0.5567, 0.2158) X_158 = -CROSS CONJUGATE OF X(92) (0.6215, 0.5117) X_3085 = INTERSECTION OF LINES X(1)X(2) AND X(4)X(12) (0.5576, 0.2197) A B C X3086 X1 X2 X4 X12 X7 X46 X20 X35 X37 X158 X3085 (a, b, c) = (5, 6, 7) | AB = 1

X3085 (INTERSECTION OF LINES X1X_2 AND X4X_12) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \(f(A,B,C) : f(B,C,A) : f(C,A,B)\), where \(f(A,B,C) = 1 + \sin B \sin C\)
Barycentrics: \((\sin A)f(A,B,C) : (\sin B)f(B,C,A) : (\sin C)f(C,A,B)\)
Search Value (6, 9, 13): \(1.929559536981\)
▸ Incident Central Lines37 lines

{1, 2}, {3, 388}, {4, 12}, {5, 497}, {7, 46}, {11, 1058}, {20, 35}, {28, 197}, {31, 3074}, {33, 3089}, {34, 3088}, {37, 158}, {40, 226}, {56, 631}, {69, 611}, {100, 377}, {140, 999}, {144, 191}, {171, 255}, {212, 3072}, {227, 278}, {238, 1497}, {346, 1089}, {390, 1479}, {405, 2551}, {42, 954}, {443, 1376}, {496, 1656}, {601, 1935}, {750, 1496}, {756, 774}, {942, 1788}, {944, 2646}, {1124, 3069}, {1335, 3068}, {1588, 2066}, {2241, 2548}

▸ Functional Relations1 relation