X3572

INTERSECTION OF LINES P(6)U(6) AND P(8)U(8)

The points given by trilinears P(6) =\(b : c : a\) and U(6) =\(c : a : b\) are indexed at Bicentric Pairs (accessible using the Tables button at the top of ETC), and likewise for their isogonal conjugates P(8) =\(1/b : 1/c : 1/a\) and U(8) =\(1/c : 1/a : 1/b\). The pair of central lines P(6)U(6) and P(8)U(8) concur in X3572. Contributed by Peter Moses, July 7, 2009.
< 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, 513 }: Central Line L(37, 513) { 2, 661 }: Central Line L(2, 661) { 42, 649 }: Central Line L(42, 649) { 6, 798 }: Central Line L(6, 798) { 111, 741 }: Central Line L(111, 741) { 335, 812 }: Central Line L(335, 812) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_3570 = INTERSECTION OF LINES P(6)U(8) AND P(8)U(6) (7.4218, 7.5984) X_37 = CROSSPOINT OF INCENTER AND CENTROID (0.5567, 0.2158) X_513 = ISOGONAL CONJUGATE OF X(100) (1.0529, -0.0806) X_2 = CENTROID (0.5374, 0.2000) X_661 = CROSSDIFFERENCE OF X(1) AND X(21) (48.4286, -23.0952) X_42 = CROSSPOINT OF INCENTER AND SYMMEDIAN POINT (0.5880, 0.2495) X_649 = CROSSDIFFERENCE OF X(1) and X(2) (-21.0000, 14.6969) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6000, 0.2672) X_798 = CROSSDIFFERENCE OF X(1) AND X(75) (-13.4299, 10.5763) X_111 = PARRY POINT (0.9885, -0.0457) X_741 = EVEN (- 1, 1) CIRCUMCIRCLE POINT (1.0071, 0.0451) X_335 = ISOTOMIC CONJUGATE OF X(239) (1.0387, 0.0318) X_812 = ODD (- 1, 1) INFINITY POINT (0.1586, -0.2483) X_3572 = INTERSECTION OF LINES P(6)U(6) AND P(8)U(8) (0.9934, -0.0218) A B C X3570 X37 X513 X2 X661 X42 X649 X6 X798 X111 X741 X335 X812 X3572 (a, b, c) = (5, 6, 7) | AB = 1

X3572 (INTERSECTION OF LINES P(6)U(6) AND P(8)U(8)) 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 \(f(a,b,c) = a(b - c)/(a^{2} - bc)\)
Barycentrics: \(af(a,b,c) : bf(b,c,a) : cf(c,a,b)\)
Search Value (6, 9, 13): \(0.075244778086\)
▸ Incident Central Lines10 lines

{2, 661}, {6, 798}, {37, 513}, {42, 649}, {111, 741}, {291, 1635}, {335, 812}, {660, 1026}, {813, 901}, {1019, 2084}

▸ Functional Relations8 relations