X2037

1st BROCARD-AXIS-APOLLONIUS-CIRCLE INTERSECTION

Of the two points of intersection of the Brocard axis and Apollonius circle, X2037 is the one nearer to X3 and also nearer to X6.
< 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) { 2038, 970 }: Brocard Axis { 3, 6 }: Brocard Axis { 10, 2039 }: Central Line L(10, 2039) { 2040, 2051 }: Central Line L(2040, 2051) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_2038 = 2nd BROCARD-AXIS-APOLLONIUS-CIRCLE INTERSECTION (-0.6835, -1.8525) X_970 = CENTER OF THE APOLLONIUS CIRCLE (0.2643, -0.2872) X_3 = CIRCUMCENTER (0.5000, 0.1021) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6000, 0.2672) X_10 = SPIEKER CENTER (0.5204, 0.1833) X_2039 = 1st NINE-POINT-CIRCLE-KIEPERT-ASYMPTOTES INTERSECTION (0.4240, 0.0306) X_2040 = = 2nd NINE-POINT-CIRCLE-KIEPERT-ASYMPTOTES INTERSECTION (0.6883, 0.4672) X_2051 = EXSIMILICENTER(NINE-POINT CIRCLE, APOLLONIUS CIRCLE) (0.6034, 0.3358) X_2037 = 1st BROCARD-AXIS-APOLLONIUS-CIRCLE INTERSECTION (1.2120, 1.2780) A B C X2038 X970 X3 X6 X10 X2039 X2040 X2051 X2037 (a, b, c) = (5, 6, 7) | AB = 1

X2037 (1st BROCARD-AXIS-APOLLONIUS-CIRCLE INTERSECTION) 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 \(f(a,b,c) = ea \cot A - (2r' - es \csc \omega ) \cos(A + \omega )\), where r’ = (\(r^{2}\) + \(s^{2})\)/(4r) = radius of Apollonius circle (where r = inradius)
Barycentrics: \(af(a,b,c) : bf(b,c,a) : cf(c,a,b)\)
Search Value (6, 9, 13): \(-4.728145784655\)
▸ Incident Central Lines3 lines

{3, 6}, {10, 2039}, {2040, 2051}

▸ Functional Relations1 relation