X1689

INVERSE-IN-BROCARD-CIRCLE OF X1687

X1689 is the external center of similitude of the Gallatly circle and the circumcircle. (Peter J. C. Moses, 9/03)
< 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) { 1690, 6 }: Brocard Axis { 3, 6 }: Brocard Axis A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_1690 = INVERSE-IN-BROCARD-CIRCLE OF X(1688) (0.5529, 0.1894) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6000, 0.2672) X_3 = CIRCUMCENTER (0.5000, 0.1021) X_1689 = INVERSE-IN-BROCARD-CIRCLE OF X(1687) (0.6471, 0.3451) A B C X1690 X6 X3 X1689 (a, b, c) = (5, 6, 7) | AB = 1

X1689 (INVERSE-IN-BROCARD-CIRCLE OF X1687) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \(\cos(A + \omega /2 + \pi /4) : \cos(B + \omega /2 + \pi /4) : \cos(C + \omega /2 + \pi /4)\)
Barycentrics: \((\sin A)f(A,B,C) : (\sin B)f(B,C,A) : (\sin C)f(C,A,B)\)
Search Value (6, 9, 13): \(-0.826180594801\)
▸ Alternate Coordinates3 forms

  • Trilinears: \(\sin(A + \omega /2 - \pi /4) : \sin(B + \omega /2 - \pi /4) : \sin(C + \omega /2 - \pi /4)\)

  • Trilinears: \(f(A,B,C) : f(B,C,A) : f(C,A,B)\), where \(f(A,B,C) = \cos(A + \omega ) - \sin A\)

  • Trilinears: \(\sin(A + \omega ) - \cos A\)

▸ Incident Central Lines11 lines

{1, 2018}, {2, 2009}, {3, 6}, {4, 2010}, {55, 2007}, {56, 2008}, {165, 2017}, {958, 2013}, {1344, 2015}, {1345, 2016}, {1376, 2014}

▸ Functional Relations4 relations