X3511

INTERSECTION OF LINES X3X_76 AND X6X_694

X3511 = crosssum of X523 and X2679
< 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) { 3, 76 }: Central Line L(3, 76) { 147, 446 }: Central Line L(147, 446) { 237, 385 }: Central Line L(237, 385) { 6, 694 }: Central Line L(6, 694) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_3 = CIRCUMCENTER (0.5000, 0.1021) X_76 = 3rd BROCARD POINT (0.4567, 0.1388) X_147 = TARRY POINT OF ANTICOMPLEMENTARY TRIANGLE (-0.1660, 1.0561) X_446 = CROSSPOINT OF X(98) AND X(511) (-1.3712, -4.7940) X_237 = -LINE CONJUGATE OF X(2) (0.9894, 1.3825) X_385 = -HIRST INVERSE OF X(6) (2.2863, 2.0795) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6000, 0.2672) X_694 = ISOGONAL CONJUGATE OF X(385) (1.0179, 0.0378) X_3511 = INTERSECTION OF LINES X(3)X(76) AND X(6)X(694) (-0.2342, 0.7251) A B C X3 X76 X147 X446 X237 X385 X6 X694 X3511 (a, b, c) = (5, 6, 7) | AB = 1

X3511 (INTERSECTION OF LINES X3X_76 AND X6X_694) 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 \(f(a,b,c) = a(a^{6}b^{4} - a^{4}b^{6} - a^{4}c^{6} + a^{6}c^{4} + b^{4}c^{6} + b^{6}c^{4} - a^{2}b^{2}c^{6} + a^{2}b^{4}c^{4} - a^{2}b^{6}c^{2} - a^{4}b^{2}c^{4} - a^{4}b^{4}c^{2} + a^{6}b^{2}c^{2})\)
Barycentrics: \(af(a,b,c) : bf(b,c,a) : cf(c,a,b)\)
Search Value (6, 9, 13): \(2.534201718067\)

X3511 lies on the cubic Z(X385).

▸ Incident Central Lines4 lines

{3, 76}, {6, 694}, {147, 446}, {237, 385}

▸ Functional Relations5 relations

Notes

X3511 is the point P on line X3X76 for which the P-Brocard triangle is perspective to ABC. (Randy Hutson, August 19, 2019)