X2116

POINT KEPLER VI

< 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, 105 }: Central Line L(6, 105) { 1423, 2107 }: Central Line L(1423, 2107) { 1931, 2115 }: Central Line L(1931, 2115) { 238, 2117 }: Central Line L(238, 2117) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6000, 0.2672) X_105 = Λ(INCENTER, SYMMEDIAN POINT) (0.9714, -0.0933) X_1423 = -BETH CONJUGATE OF X(192) (0.6537, 0.3692) X_2107 = -ISOCONJUGATE OF X(2106) (1.0304, 0.0996) X_1931 = CROSSDIFFERENCE OF PU(32) (1.7418, 1.7715) X_2115 = -ISOCONJUGATE OF X(2114) (-0.5439, 0.4629) X_238 = -LINE CONJUGATE OF X(37) (1.6190, 1.4775) X_2117 = -ISOCONJUGATE OF X(2116) (-0.4662, 0.5808) X_2116 = POINT KEPLER VI (0.0488, 0.8023) A B C X6 X105 X1423 X2107 X1931 X2115 X238 X2117 X2116 (a, b, c) = (5, 6, 7) | AB = 1

X2116 (POINT KEPLER VI) 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 \(f(a,b,c) = u(b,c,a)u(c,a,b)(d_{1}a^{5} - d_{2}a^{4} - d_{3}a^{3} - d_{4}a^{2} - d_{5}a - d_{6})\), where \(u(a,b,c) = a^{2} - 2bc - ac - bc\), \(d_{1}(a,b,c) = 2(b + c)\), \(d_{2}(a,b,c) = b^{2} - bc + c^{2}\), \(d_{3}(a,b,c) = 3bc(b + c)\), \(d_{4}(a,b,c) = b^{4} + b^{2}c^{2} + c^{4} - b^{3}c - c^{3}b\), \(d_{5}(a,b,c) = bc(b + c)(b^{2} - 4bc + c^{2})\), \(d_{6}(a,b,c) = b^{2}c^{2}(b - c)^{2}\)
Barycentrics: \(af(a,b,c) : bf(b,c,a) : cf(c,a,b)\)
Search Value (6, 9, 13): \(0.391783809558\)
▸ Incident Central Lines4 lines

{6, 105}, {238, 2117}, {1423, 2107}, {1931, 2115}