X3363

X6 OF PEDAL TRIANGLE OF X2

X3363 is the symmedian point of the pedal triangle of the centroid. A second construction uses the midpoint X597 of the symmedian point and centroid and the midpoint X115 of the two Fermat points, X13 and X14; specifically, X3363 is the point in which the line X115X597 meets the Euler line. Contributed by Po-chieh Chen and Shao-cheng Liu, May 29, 2008.
< 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) { 2, 3 }: Euler Line { 115, 597 }: Central Line L(115, 597) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_2 = CENTROID (0.5374, 0.2000) X_3 = CIRCUMCENTER (0.5000, 0.1021) X_4 = ORTHOCENTER (0.6122, 0.3958) X_115 = CENTER OF KIEPERT HYPERBOLA (0.7507, 0.0838) X_597 = MIDPOINT OF X(2) AND X(6) (0.5687, 0.2336) X_3363 = OF PEDAL TRIANGLE OF X(2) (0.5547, 0.2451) A B C X2 X3 X4 X115 X597 X3363 (a, b, c) = (5, 6, 7) | AB = 1

X3363 (X6 OF PEDAL TRIANGLE OF X2) 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 \(f(a,b,c) = bc(-4a^{4} + 5b^{4} + 5c^{4} - 14b^{2}c^{2} - 5c^{2}a^{2} - 5a^{2}b^{2})\)
Barycentrics: \(af(a,b,c) : bf(b,c,a) : cf(c,a,b)\)
Search Value (6, 9, 13): \(1.602164244531\)
▸ Incident Central Lines2 lines

{2, 3}, {115, 597}

Notes

As a point on the Euler line, X3363 has Shinagawa coefficients \((2(E + F)^{2} + 3S^{2},9S^{2})\).