X2051

EXSIMILICENTER(NINE-POINT CIRCLE, APOLLONIUS CIRCLE)

X2051 is the external center of similitude of the nine-point and Apollonius circles. Trilinears for the internal center, X10, result from g(a,b,c) by changing the “-” just after “cos A″ to “+”. (The two circles are described just before X1662.)
< A B C >: Reference Triangle △ABC < M_a M_b M_c >: Medial Triangle △M_a M_b M_c [ A B ]: Sideline AB (Base c = 1) [ B C ]: Sideline BC (Side a) [ C A ]: Sideline CA (Side b) { 2, 1764 }: Central Line L(2, 1764) { 5, 10 }: Central Line L(5, 10) { 11, 181 }: Central Line L(11, 181) { 27, 275 }: Central Line L(27, 275) { 4, 386 }: Central Line L(4, 386) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) M_a = Midpoint M_a of BC (0.8061, 0.2999) M_b = Midpoint M_b of CA (0.3061, 0.2999) M_c = Midpoint M_c of AB (0.5000, 0.0000) X_2 = CENTROID (0.5374, 0.2000) X_1764 = MIMOSA TRANSFORM OF X(332) (0.4054, -0.0717) X_5 = NINE-POINT CENTER (0.5561, 0.2489) X_10 = SPIEKER CENTER (0.5204, 0.1833) X_11 = FEUERBACH POINT (0.7347, 0.0667) X_181 = APOLLONIUS POINT (0.6163, 0.3093) X_27 = CEVAPOINT OF ORTHOCENTER AND CLAWSON CENTER (0.6182, 0.4112) X_275 = CEVAPOINT OF ORTHOCENTER AND SYMMEDIAN POINT (0.6057, 0.3473) X_4 = ORTHOCENTER (0.6122, 0.3958) X_386 = INVERSE-IN-BROCARD-CIRCLE OF X(58) (0.5912, 0.2526) X_2051 = EXSIMILICENTER(NINE-POINT CIRCLE, APOLLONIUS CIRCLE) (0.6034, 0.3358) A B C Ma Mb Mc X2 X1764 X5 X10 X11 X181 X27 X275 X4 X386 X2051 (a, b, c) = (5, 6, 7) | AB = 1

X2051 (EXSIMILICENTER(NINE-POINT CIRCLE, APOLLONIUS CIRCLE)) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: (\(s^{2}\) - \(r^{2})\)cos A - (\(s^{2}\) + \(r^{2})\)cos(B - C) - 2rs sin A (Peter J. C. Moses)
Barycentrics: \(1/[a^{3} - a(b^{2} - bc + c^{2}) - bc(b + c)] (Paul Yiu)\)
Search Value (6, 9, 13): \(-2.727767488977\)
▸ Alternate Coordinates3 forms

  • Trilinears: \(\sec(A - U) : :\), U as at X572

  • Trilinears: \(1/(s \cos A + r \sin A) : :\)

  • Barycentrics: \(1/(a^{3} - a (b^{2} - b c + c^{2}) - b c(b + c)) : :\)

▸ Incident Central Lines23 lines

{2, 573}, {4, 386}, {5, 10}, {6, 2050}, {11, 181}, {12, 1682}, {27, 275}, {43, 1699}, {226, 1465}, {321, 908}, {469, 2052}, {485, 1685}, {486, 1686}, {572, 2185}, {1348, 1693}, {1349, 1694}, {1676, 1683}, {1677, 1684}, {1695, 1698}, {1766, 2339}, {2009, 2019}, {2101, 2020}, {2037, 2040}

▸ Functional Relations6 relations

Notes

Let A' be the midpoint of BC, A″ the midpoint of AX1, and define B', C' and B″, C″ cyclically. Let A“‘B“‘C”’ be the incentral triangle. Let A* be the orthocenter of A’A”A”’, and define B* and C* cyclically. The lines AA*, BB*, CC* concur in X2051. (Randy Hutson, March 21, 2019)

Let LA be the radical axis of the A-excircle and the excircles radical circle, and define LB and LC cyclically. Let A' = LB∩LC, and define B' and C' cyclically. The lines AA', BB', CC' concur in X2051. (Randy Hutson, March 21, 2019)

Let A'B'C' be the tangential triangle of the Feuerbach triangle. Let A″B″C″ be the tangential triangle of the Apollonius triangle. Let A* be the {A',A″}-harmonic conjugate of X10, and define B* and C* cyclically. The lines AA*, BB*, CC* concur in X2051. (Randy Hutson, March 21, 2019)