X1471

X1001-BETH CONJUGATE OF X1001

Let A1B1C1 and A2B2C2 be the 1st and 2nd Auriga triangles. Let A' be the trilinear product A1*A2, and define B', C' cyclically. The lines AA', BB', CC' concur in X1471. (Randy Hutson, March 21, 2019)
< 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, 41 }: Central Line L(6, 41) { 31, 57 }: Central Line L(31, 57) { 7, 238 }: Central Line L(7, 238) { 58, 269 }: Central Line L(58, 269) { 212, 354 }: Central Line L(212, 354) { 226, 748 }: Central Line L(226, 748) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_1001 = MIDPOINT OF X(1) AND X(9) (0.5412, 0.1974) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6000, 0.2672) X_41 = -CEVA CONJUGATE OF X(31) (0.5823, 0.2244) X_31 = 2nd POWER POINT (0.6228, 0.3008) X_57 = ISOGONAL CONJUGATE OF X(9) (0.6138, 0.3110) X_7 = GERGONNE POINT (0.5903, 0.2769) X_238 = -LINE CONJUGATE OF X(37) (1.6190, 1.4775) X_58 = ISOGONAL CONJUGATE OF X(10) (0.6107, 0.2850) X_269 = ISOGONAL CONJUGATE OF X(200) (0.6369, 0.3847) X_212 = -CEVA CONJUGATE OF X(41) (0.4719, 0.0786) X_354 = WEILL POINT (0.5823, 0.2533) X_226 = -CEVA CONJUGATE OF X(65) (0.5778, 0.2588) X_748 = INTERSECTION OF LINES X(2)X(31) AND X(9)X(38) (0.4360, 0.0802) X_1471 = -BETH CONJUGATE OF X(1001) (0.6167, 0.3077) A B C X1001 X6 X41 X31 X57 X7 X238 X58 X269 X212 X354 X226 X748 X1471 (a, b, c) = (5, 6, 7) | AB = 1

X1471 (X1001-BETH CONJUGATE OF X1001) 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 \(f(a,b,c) = (1 - \cos A)u(a,b,c)\), where \(u : v : w\) = X1001
Barycentrics: \(af(a,b,c) : bf(b,c,a) : cf(c,a,b)\)
Search Value (6, 9, 13): \(0.416622373242\)
▸ Incident Central Lines13 lines

{1, 1170}, {6, 41}, {7, 238}, {31, 57}, {36, 991}, {58, 269}, {65, 1279}, {212, 354}, {226, 748}, {241, 1386}, {307, 1125}, {602, 942}, {603, 1418}