X1336

ISOGONAL CONJUGATE OF X1335

X1336 is the isogonal conjugate of X1335.
< 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) { 37, 158 }: Central Line L(37, 158) { 57, 481 }: Central Line L(57, 481) { 2, 585 }: Central Line L(2, 585) { 81, 1124 }: Central Line L(81, 1124) { 274, 1267 }: Central Line L(274, 1267) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_1335 = {X(1),X(6)}-HARMONIC CONJUGATE OF X(1124) (0.4024, 0.0326) X_37 = CROSSPOINT OF INCENTER AND CENTROID (0.5567, 0.2158) X_158 = -CROSS CONJUGATE OF X(92) (0.6215, 0.5117) X_57 = ISOGONAL CONJUGATE OF X(9) (0.6138, 0.3110) X_481 = 1st EPPSTEIN POINT (0.6148, 0.3335) X_2 = CENTROID (0.5374, 0.2000) X_585 = 1st CONGRUENT SHRUNK INSQUARES POINT (0.5354, 0.1924) X_81 = CEVAPOINT OF INCENTER AND SYMMEDIAN POINT (0.5849, 0.2510) X_1124 = ISOGONAL CONJUGATE OF X(1123) (0.5846, 0.2489) X_274 = ISOGONAL CONJUGATE OF X(213) (0.5168, 0.1843) X_1267 = ISOTOMIC CONJUGATE OF X(1123) (0.5162, 0.1825) X_1336 = ISOGONAL CONJUGATE OF X(1335) (0.6230, 0.5188) A B C X1335 X37 X158 X57 X481 X2 X585 X81 X1124 X274 X1267 X1336 (a, b, c) = (5, 6, 7) | AB = 1

X1336 (ISOGONAL CONJUGATE OF X1335) 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 \(f(A,B,C) = 1/(1 - \sin A)\)
Barycentrics: \((\sin A)f(A,B,C) : (\sin B)f(B,C,A) : (\sin C)f(C,A,B)\)
Search Value (6, 9, 13): \(0.575148918097\)
▸ Incident Central Lines9 lines

{1, 3068}, {2, 585}, {4, 2362}, {37, 158}, {57, 481}, {81, 1124}, {274, 1267}, {498, 3300}, {499, 3302}

▸ Functional Relations3 relations

Notes

The parabola with focus A and directrix BC meets line AB in two points; let AB be the one further from B, and define AC similarly. Let LA be the line ABAC, and define LB and LC cyclically. Let A' = LB∩LC, and define B' and C' cyclically. The lines AA', BB', CC' concur in X1336. (Randy Hutson, 9/23/2011)

In the configuration for the Paasche point X1123, there are 4 circles tangent to the circle with diameter BC and also tangent to the lines AB and AC. Of the 4 circles, there are two pairs, one having X1123 as perspector, and the other having X1336. (Peter Moses, 21 January 2013)