X2009

INSIMILICENTER(GALLATLY CIRCLE, NINE-POINT CIRCLE)

X2009 is the insimilicenter(gallatly circle, nine-point circle).
< 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) { 2010, 115 }: Central Line L(2010, 115) { 5, 39 }: Central Line L(5, 39) { 2, 1689 }: Central Line L(2, 1689) { 4, 1690 }: Central Line L(4, 1690) { 83, 1688 }: Central Line L(83, 1688) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_2010 = EXSIMILICENTER(GALLATLY CIRCLE, NINE-POINT CIRCLE) (0.9341, -0.0718) X_115 = CENTER OF KIEPERT HYPERBOLA (0.7507, 0.0838) X_5 = NINE-POINT CENTER (0.5561, 0.2489) X_39 = BROCARD MIDPOINT (0.5778, 0.2305) X_2 = CENTROID (0.5374, 0.2000) X_1689 = INVERSE-IN-BROCARD-CIRCLE OF X(1687) (0.6471, 0.3451) X_4 = ORTHOCENTER (0.6122, 0.3958) X_1690 = INVERSE-IN-BROCARD-CIRCLE OF X(1688) (0.5529, 0.1894) X_83 = CEVAPOINT OF CENTROID AND SYMMEDIAN POINT (0.5651, 0.2360) X_1688 = EXSIMILICENTER(1st LEMOINE CIRCLE, 2nd LEMOINE CIRCLE) (1.4224, 1.6255) X_2009 = INSIMILICENTER(GALLATLY CIRCLE, NINE-POINT CIRCLE) (0.5673, 0.2394) A B C X2010 X115 X5 X39 X2 X1689 X4 X1690 X83 X1688 X2009 (a, b, c) = (5, 6, 7) | AB = 1

X2009 (INSIMILICENTER(GALLATLY CIRCLE, NINE-POINT CIRCLE)) 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 \(f(A,B,C) = \cos(B - C) + \sin(A + \omega )\)
Barycentrics: \((\sin A)f(A,B,C) : (\sin B)f(B,C,A) : (\sin C)f(C,A,B)\)
Search Value (6, 9, 13): \(1.174094279298\)
▸ Alternate Coordinates1 form

  • Trilinears: \(h(A,B,C) : h(B,C,A): h(C,A,B)\), where \(h(A,B,C) = \csc(A - \omega /2 + \pi /4)\) (M. Iliev, 5/13/07)

X2009 lies on the Kiepert hyperbola.

▸ Incident Central Lines18 lines

{2, 1689}, {4, 1690}, {5, 39}, {10, 2020}, {11, 2007}, {12, 2008}, {83, 1688}, {98, 1687}, {485, 1670}, {486, 1671}, {1346, 2016}, {1347, 2015}, {1348, 2012}, {1349, 2011}, {1698, 2018}, {1699, 2017}, {2026, 2040}, {2027, 2039}

▸ Functional Relations3 relations