X3448

ISOGONAL CONJUGATE OF X3447

X3448 is the isogonal conjugate of X3447.
< 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) { 20, 68 }: Central Line L(20, 68) { 4, 94 }: Central Line L(4, 94) { 2, 98 }: Central Line L(2, 98) { 67, 69 }: Central Line L(67, 69) { 66, 193 }: Central Line L(66, 193) { 5, 399 }: Central Line L(5, 399) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_3447 = -VERTEX CONJUGATE OF X(523) (-0.1724, -0.4378) X_20 = DE LONGCHAMPS POINT (0.3878, -0.1916) X_68 = PRASOLOV POINT (-0.0535, -0.0055) X_4 = ORTHOCENTER (0.6122, 0.3958) X_94 = ISOGONAL CONJUGATE OF X(50) (0.9069, -0.1331) X_2 = CENTROID (0.5374, 0.2000) X_98 = TARRY POINT (0.8891, -0.2281) X_67 = ISOGONAL CONJUGATE OF X(23) (0.7606, -0.2152) X_69 = SYMMEDIAN POINT OF THE ANTICOMPLEMENTARY TRIANGLE (0.4122, 0.0654) X_66 = ISOGONAL CONJUGATE OF X(22) (0.5912, 1.0593) X_193 = -CEVA CONJUGATE OF X(2) (0.7878, 0.4690) X_5 = NINE-POINT CENTER (0.5561, 0.2489) X_399 = PARRY REFLECTION POINT (0.0032, 0.9936) X_3448 = ISOGONAL CONJUGATE OF X(3447) (1.1090, -0.4958) A B C X3447 X20 X68 X4 X94 X2 X98 X67 X69 X66 X193 X5 X399 X3448 (a, b, c) = (5, 6, 7) | AB = 1

X3448 (ISOGONAL CONJUGATE OF X3447) 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 \(f(a,b,c) = 1/x(3447)\)
Search Value (6, 9, 13): \(6.072980429263\)
▸ Alternate Coordinates1 form

  • Trilinears: \(g(a,b,c) : g(b,c,a) : g(c,a,b)\), where \(g(a,b,c) = bc[a^{6} - a^{4}(b^{2} + c^{2}) + a^{2}(b^{4} - b^{2}c^{2} + c^{4}) - (b^{2} + c^{2})(b^{2} - c^{2})^{2}] (E\). Danneels)

X3448 lies on the anticomplementary circle.

▸ Incident Central Lines23 lines

{2, 98}, {3, 2888}, {4, 94}, {5, 399}, {10, 2948}, {20, 68}, {23, 1503}, {66, 193}, {67, 69}, {141, 2930}, {148, 690}, {150, 2774}, {151, 2779}, {152, 2772}, {153, 355}, {246, 2782}, {323, 858}, {427, 1353}, {631, 1511}, {1330, 2842}, {1853, 1993}, {2836, 2895}, {2918, 2931}

▸ Functional Relations10 relations

Notes

Let A'B'C' be the orthocentroidal triangle and A″B″C″ the anti-orthocentroidal triangle. Let A* be the reflection of A″ in B'C', and define B* and C* cyclically. The lines AA*, BB*, CC* concur in X74, and X3448 = centroid of A*B*C*. (Randy Hutson, December 10, 2016)