X1988

2nd LEMOINE ANTIPARALLELS POINT

X1988 is discussed in Lemoine’s paper cited at X19. Contributed by Darij Grinberg.
< 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) { 6, 436 }: Central Line L(6, 436) { 394, 401 }: Central Line L(394, 401) { 184, 1968 }: Central Line L(184, 1968) { 577, 1971 }: Central Line L(577, 1971) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_4 = ORTHOCENTER (0.6122, 0.3958) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6000, 0.2672) X_436 = EULER X(48)-7th-SUBSTITUTION POINT (0.6027, 0.3709) X_394 = -CEVA CONJUGATE OF X(3) (0.3778, 0.0284) X_401 = BAILEY POINT (-0.0758, -1.4044) X_184 = INVERSE OF X(125) IN THE BROCARD CIRCLE (0.5788, 0.1496) X_1968 = CROSSSUM OF PU(37) (0.5947, 0.4354) X_577 = HARMONIC OF X(216) (0.4663, 0.0464) X_1971 = CROSSDIFFERENCE OF PU(38) (1.3141, 4.2335) X_1988 = 2nd LEMOINE ANTIPARALLELS POINT (0.6134, 0.7727) A B C X4 X6 X436 X394 X401 X184 X1968 X577 X1971 X1988 (a, b, c) = (5, 6, 7) | AB = 1

X1988 (2nd LEMOINE ANTIPARALLELS POINT) 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 \(f(A,B,C) = (\sin A)/(\csc 2B + \csc 2C - \csc 2A)\)
Barycentrics: \((\sin A)f(A,B,C) : (\sin B)f(B,C,A) : (\sin C)f(C,A,B)\)
Search Value (6, 9, 13): \(2.712216602766\)
▸ Incident Central Lines4 lines

{6, 436}, {184, 1968}, {394, 401}, {577, 1971}

▸ Functional Relations2 relations