X3612

INTERSECTION OF LINES X1X_3 AND X21X_90

Let Ja, Jb, Jc be the excenters and I the incenter. Let A' be the centroid of JbJcI, and define B' and C' cyclically. A'B'C' is also the cross-triangle of the excentral and 2nd circumperp triangles. A'B'C' is homothetic to the reflection triangle of X1 at X3612. (Randy Hutson, July 31 2018)
< 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, 3 }: OI Line (Incenter–Circumcenter Line) { 21, 90 }: Central Line L(21, 90) { 498, 515 }: Central Line L(498, 515) { 78, 993 }: Central Line L(78, 993) { 499, 950 }: Central Line L(499, 950) { 377, 1125 }: Central Line L(377, 1125) 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_3 = CIRCUMCENTER (0.5000, 0.1021) X_21 = SCHIFFLER POINT (0.5287, 0.1771) X_90 = -CROSS CONJUGATE OF X(1) (0.1022, 0.0201) X_498 = YFF CONCURRENT CONGRUENT CIRCLES POINT (0.5518, 0.2140) X_515 = ISOGONAL CONJUGATE OF X(102) (0.6804, 0.7691) X_78 = ISOGONAL CONJUGATE OF X(34) (0.3835, 0.0491) X_993 = MIDPOINT OF X(1) AND X(63) (0.5140, 0.1578) X_499 = {X(1),X(2)}-HARMONIC CONJUGATE OF X(498) (0.4458, 0.1102) X_950 = CROSSPOINT OF X(8) AND X(29) (0.5621, 0.1961) X_377 = EULER LINE INTERCEPT OF LINE X(7)X(8) (0.5473, 0.2258) X_1125 = COMPLEMENT OF X(10) (0.5459, 0.2083) X_3612 = INTERSECTION OF LINES X(1)X(3) AND X(21)X(90) (0.5439, 0.1827) A B C X1 X3 X21 X90 X498 X515 X78 X993 X499 X950 X377 X1125 X3612 (a, b, c) = (5, 6, 7) | AB = 1

X3612 (INTERSECTION OF LINES X1X_3 AND X21X_90) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \(3 \cos A + \cos B + \cos C : :\)
Search Value (6, 9, 13): \(3.068868072161\)
▸ Incident Central Lines21 lines

{1, 3}, {9, 2174}, {21, 90}, {78, 993}, {80, 1698}, {140, 1837}, {186, 1452}, {284, 1723}, {376, 1770}, {377, 1125}, {442, 3586}, {498, 515}, {499, 950}, {550, 1836}, {631, 1737}, {1006, 1728}, {1047, 3362}, {1449, 2245}, {1788, 3524}, {1905, 3515}, {3474, 3528}

▸ Functional Relations1 relation