X347

ISOTOMIC CONJUGATE OF X280

X347 is the isotomic conjugate of X280.
< 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, 7 }: Soddy Line { 2, 92 }: Central Line L(2, 92) { 8, 253 }: Central Line L(8, 253) { 75, 280 }: Central Line L(75, 280) { 144, 219 }: Central Line L(144, 219) { 227, 322 }: Central Line L(227, 322) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_280 = -CROSS CONJUGATE OF X(8) (0.6379, 0.8041) X_1 = INCENTER (0.5714, 0.2333) X_7 = GERGONNE POINT (0.5903, 0.2769) X_2 = CENTROID (0.5374, 0.2000) X_92 = CEVAPOINT OF INCENTER AND CLAWSON POINT (0.5949, 0.3639) X_8 = NAGEL POINT (0.4694, 0.1333) X_253 = -CROSS CONJUGATE OF X(2) (3.5087, 32.5688) X_75 = ISOTOMIC CONJUGATE OF INCENTER (0.4988, 0.1682) X_144 = ANTICOMPLEMENT OF X(7) (0.4317, 0.0461) X_219 = -CEVA CONJUGATE OF X(55) (0.4277, 0.0625) X_227 = -CEVA CONJUGATE OF X(65) (0.4792, -0.0460) X_322 = ISOTOMIC CONJUGATE OF X(84) (0.3430, -0.0168) X_347 = ISOTOMIC CONJUGATE OF X(280) (0.4531, -0.0404) A B C X280 X1 X7 X2 X92 X8 X253 X75 X144 X219 X227 X322 X347 (a, b, c) = (5, 6, 7) | AB = 1

X347 (ISOTOMIC CONJUGATE OF X280) 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 \(f(A,B,C) = (-1 - \cos A + \cos B + \cos C) \sec^{2}(A/2)\)
Barycentrics: \(af(A,B,C) : bf(B,C,A) : cf(C,A,B)\)
Search Value (6, 9, 13): \(-2.197658032984\)
▸ Incident Central Lines12 lines

{1, 7}, {2, 92}, {8, 253}, {34, 452}, {37, 948}, {69, 664}, {75, 280}, {144, 219}, {223, 329}, {227, 322}, {241, 1108}, {573, 1020}

▸ Functional Relations10 relations

Notes

Let Ha be the hyperbola passing through A, and with foci at B and C. This is called the A-Soddy hyperbola in (Paul Yiu, Introduction to the Geometry of the Triangle, 2002; 12.4 The Soddy hyperbolas, p. 143). Let La be the polar of X20 with respect to Ha. Define Lb, Lc cyclically. La, Lb, Lc concur in X347. (Randy Hutson, November 30, 2018)