X2092

DANNEELS-APOLLONIUS PERSPECTOR

Let A' be the point common to Apollonius circle and the A-excircle, and define B' and C' cyclically. Then triangle A'B'C' is perspective to the cevian triangle of X6, and the perspector is X2092. (Eric Danneels, Hyacinthos, #8070, 10/01/03). (Triangle A'B'C' is also perspective to the cevian triangle of X1; see X1682.)
< A B C >: Reference Triangle △ABC < P_a P_b P_c >: Cevian Triangle △P_a P_b P_c [ A B ]: Sideline AB (Base c = 1) [ B C ]: Sideline BC (Side a) [ C A ]: Sideline CA (Side b) [ A P_a ]: A-Cevian A–P_a [ B P_b ]: B-Cevian B–P_b [ C P_c ]: C-Cevian C–P_c { 3, 6 }: Brocard Axis { 10, 37 }: Central Line L(10, 37) { 9, 43 }: Central Line L(9, 43) { 42, 181 }: Central Line L(42, 181) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) P_a = Cevian Trace P_a on BC (0.7902, 0.3246) P_b = Cevian Trace P_b on CA (0.3698, 0.3624) P_c = Cevian Trace P_c on AB (0.5640, 0.0000) 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_37 = CROSSPOINT OF INCENTER AND CENTROID (0.5567, 0.2158) X_9 = MITTENPUNKT (0.5110, 0.1615) X_43 = -CEVA CONJUGATE OF X(1) (0.6358, 0.2963) X_42 = CROSSPOINT OF INCENTER AND SYMMEDIAN POINT (0.5880, 0.2495) X_181 = APOLLONIUS POINT (0.6163, 0.3093) X_2092 = DANNEELS-APOLLONIUS PERSPECTOR (0.5833, 0.2396) A B C Pa Pb Pc X3 X6 X10 X37 X9 X43 X42 X181 X2092 (a, b, c) = (5, 6, 7) | AB = 1

X2092 (DANNEELS-APOLLONIUS PERSPECTOR) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

  • Derived Triangle: Cevian Triangle △PaPbPc — vertices Pa, Pb, Pc (Traces of cevians through target center on BC, CA, AB)

  • Concurrence Lines: A–Pa, B–Pb, C–Pc concurring at X2092

Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\), where \(f(a,b,c) = a^{2}(b + c)^{2} + a(b + c)(b^{2} + c^{2})\)
Barycentrics: \(af(a,b,c) : bf(b,c,a) : cf(c,a,b)\)
Search Value (6, 9, 13): \(1.427222829750\)
▸ Incident Central Lines53 lines

{1, 2277}, {2, 314}, {3, 6}, {4, 3597}, {9, 43}, {10, 37}, {41, 2273}, {42, 181}, {44, 3647}, {69, 980}, {71, 213}, {100, 2298}, {115, 119}, {142, 3752}, {192, 3596}, {214, 1015}, {232, 1172}, {291, 2663}, {407, 1880}, {442, 1738}, {538, 3770}, {656, 3126}, {665, 2642}, {851, 2285}, {869, 3688}, {900, 2511}, {966, 5283}, {968, 4204}, {992, 3216}, {1017, 3031}, {1107, 3686}, {1193, 1682}, {1194, 5276}, {1196, 5275}, {1211, 3666}, {1449, 2275}, {2171, 4642}, {2178, 2242}, {2183, 2653}, {2240, 5279}, {2294, 3125}, {2303, 5277}, {2309, 3271}, {2610, 3310}, {2667, 3122}, {3030, 3124}, {3136, 3914}, {3247, 4050}, {3454, 3821}, {3662, 3936}, {3728, 4111}, {3949, 3954}, {4016, 4053}

▸ Functional Relations10 relations