X3345

ISOGONAL CONJUGATE OF X1490

X3345 is the isogonal conjugate of X1490.
< 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) { 3182, 3 }: Central Line L(3182, 3) { 20, 78 }: Central Line L(20, 78) { 1, 196 }: Central Line L(1, 196) { 3, 223 }: Central Line L(3, 223) { 40, 219 }: Central Line L(40, 219) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_3182 = -CEVA CONJUGATE OF X(1) (0.5164, 5.7648) X_3 = CIRCUMCENTER (0.5000, 0.1021) X_20 = DE LONGCHAMPS POINT (0.3878, -0.1916) X_78 = ISOGONAL CONJUGATE OF X(34) (0.3835, 0.0491) X_1 = INCENTER (0.5714, 0.2333) X_196 = -CEVA CONJUGATE OF X(4) (0.5649, -0.1989) X_223 = -CEVA CONJUGATE OF X(57) (0.4996, -0.0528) X_40 = BEVAN POINT (0.4286, -0.0292) X_219 = -CEVA CONJUGATE OF X(55) (0.4277, 0.0625) X_3345 = ISOGONAL CONJUGATE OF X(1490) (0.4836, -5.5606) A B C X3182 X3 X20 X78 X1 X196 X223 X40 X219 X3345 (a, b, c) = (5, 6, 7) | AB = 1

X3345 (ISOGONAL CONJUGATE OF X1490) 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 \(f(A,B,C) = 1/(1 + \sec A - \sec B - \sec C)\)
Barycentrics: \((\sin A)f(A,B,C) : (\sin B)f(B,C,A) : (\sin C)f(C,A,B)\)
Search Value (6, 9, 13): \(15.553222644323\)

X3345 lies on the Darboux cubic.

▸ Incident Central Lines11 lines

{1, 196}, {3, 223}, {4, 282}, {20, 78}, {40, 219}, {64, 3472}, {84, 2130}, {165, 1794}, {283, 1817}, {1722, 2636}, {1753, 2338}

▸ Functional Relations7 relations

Notes

Let A'B'C' be the cevian triangle of X329. Let A″ be the orthocenter of AB'C', and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X3345. (Randy Hutson, July 11, 2019)