X2114

POINT KEPLER V

< 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) { 9, 77 }: Central Line L(9, 77) { 31, 57 }: Central Line L(31, 57) { 238, 241 }: Central Line L(238, 241) { 269, 292 }: Central Line L(269, 292) { 1279, 1429 }: Central Line L(1279, 1429) { 1458, 2113 }: Central Line L(1458, 2113) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_9 = MITTENPUNKT (0.5110, 0.1615) X_77 = ISOGONAL CONJUGATE OF X(33) (0.5270, 0.1305) X_31 = 2nd POWER POINT (0.6228, 0.3008) X_57 = ISOGONAL CONJUGATE OF X(9) (0.6138, 0.3110) X_238 = -LINE CONJUGATE OF X(37) (1.6190, 1.4775) X_241 = -LINE CONJUGATE OF X(55) (1.1217, 1.2442) X_269 = ISOGONAL CONJUGATE OF X(200) (0.6369, 0.3847) X_292 = -HIRST INVERSE OF X(291) (1.0127, 0.0417) X_1279 = MIDPOINT OF X(1) AND X(238) (1.0952, 0.8554) X_1429 = -BETH CONJUGATE OF X(239) (0.9040, 0.8443) X_1458 = -BETH CONJUGATE OF X(518) (0.8889, 0.9677) X_2113 = -ISOCONJUGATE OF X(2112) (-0.3440, 0.6801) X_2114 = POINT KEPLER V (0.1793, 0.8022) A B C X9 X77 X31 X57 X238 X241 X269 X292 X1279 X1429 X1458 X2113 X2114 (a, b, c) = (5, 6, 7) | AB = 1

X2114 (POINT KEPLER V) 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 \(f(a,b,c) = (c + a - b)(a + b - c)[a^{4} + (b + c)a^{3} - u(a,b,c)a^{2} + v(a,b,c)a - w(a,b,c)]\), where \(u(a,b,c) = 2b^{2} + 3bc + 2c^{2}\), \(v(a,b,c) = (b + c)(b^{2} + c^{2})\), \(w(a,b,c) = (b - c)^{2}(b^{2} + bc + c^{2})\)
Barycentrics: \(af(a,b,c) : bf(b,c,a) : cf(c,a,b)\)
Search Value (6, 9, 13): \(0.079434058253\)
▸ Incident Central Lines7 lines

{9, 77}, {31, 57}, {238, 241}, {269, 292}, {1279, 1429}, {1458, 2113}, {2106, 2117}