X353

INVERSE-IN-BROCARD-CIRCLE OF X111

Let AaBaCa, AbBbCb, AcBcCc be the A-, B- and C-anti-altimedial triangles. Let A' be the trilinear product Aa*Ab*Ac, and define B' and C' cyclically. Triangle A'B'C' is the anticomplementary triangle of the 1st Brocard triangle, and is similar to the anti-orthocentroidal triangle, with similitude center X353. (Randy Hutson, November 2, 2017)
< 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) circumcircle: Circumcircle of △ABC { 3, 352 }: Central Line L(3, 352) { 6, 23 }: Central Line L(6, 23) { 111, 182 }: Central Line L(111, 182) { 110, 574 }: Central Line L(110, 574) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_3 = CIRCUMCENTER (0.5000, 0.1021) X_352 = INVERSE-IN-CIRCUMCIRCLE OF X(353) (2.2545, 2.0453) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6000, 0.2672) X_23 = FAR-OUT POINT (1.3871, 2.4232) X_111 = PARRY POINT (0.9885, -0.0457) X_182 = MIDPOINT OF BROCARD DIAMETER (0.5500, 0.1846) X_110 = FOCUS OF KIEPERT PARABOLA (0.2516, 0.5478) X_574 = HARMONIC OF X(187) (0.5539, 0.1910) X_353 = INVERSE-IN-BROCARD-CIRCLE OF X(111) (0.5667, 0.1759) A B C X3 X352 X6 X23 X111 X182 X110 X574 X353 (a, b, c) = (5, 6, 7) | AB = 1

X353 (INVERSE-IN-BROCARD-CIRCLE OF X111) 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 \(f(a,b,c) = a(4a^{4} - 2b^{4} - 2c^{4} - b^{2}c^{2} - 4a^{2}b^{2} -4a^{2}c^{2})\)
Barycentrics: \(af(a,b,c) : bf(b,c,a) : cf(c,a,b)\)
Search Value (6, 9, 13): \(3.283953007817\)

X353 lies on the Parry circle.

▸ Incident Central Lines5 lines

{3, 352}, {6, 23}, {110, 574}, {111, 182}, {187, 3117}

▸ Functional Relations7 relations