X₁₈₁

APOLLONIUS POINT

Let O(A),O(B),O(C) be the excircles. Apollonius’s Problem includes the construction of the circle O tangent to the three excircles and encompassing them. (The circle is called the Apollonius circle.) Let A' = O∩O(A), B'=O∩O(B), C'=O∩O(C). The lines AA',BB',CC' concur in X₁₈₁. Yff derived trilinears in 1992.

5, 6, 7 3, 4, 5 Kepler Kimberling

X₁₈₁ (APOLLONIUS POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {10, 12}, {31, 51}

• Related Centers: X₂₆₁ (ISOTOMIC CONJUGATE OF X₁₂), X₁₀ (SPIEKER CENTER), X₁₂ ({X), X₃₁ (2nd POWER POINT), X₅₁ (CENTROID OF ORTHIC TRIANGLE)

Trilinears: \(a(b + c)^{2}/(b + c - a) : b(c + a)^{2}/(c + a - b) : c(a + b)^{2}/(a + b - c)\)

Barycentrics: \(a^{3}\cos^{2}(B/2 - C/2) : b^{3}\cos^{2}(C/2 - A/2) : c^{3}\cos^{2}(A/2 - B/2)\)

Search Value (6, 9, 13): \(0.373069446401\)

▸ Alternate Coordinates2 forms

• Trilinears: \(a^{2}\cos^{2}(B/2 - C/2) : b^{2}\cos^{2}(C/2 - A/2) : c^{2}\cos^{2}(A/2 - B/2)\)

• Trilinears: \(h(a,b,c) : h(b,c,a) : h(c,a,b)\), where \(h(a,b,c) = [r \cos(A/2) + s \sin(A/2)]^{2}\), s = semiperimeter, r = inradius

▸ Incident Central Lines55 lines

{1, 970}, {6, 197}, {8, 959}, {10, 12}, {11, 2051}, {25, 2175}, {31, 51}, {33, 3022}, {42, 228}, {43, 57}, {44, 375}, {55, 573}, {56, 386}, {58, 1324}, {81, 5061}, {171, 511}, {182, 5329}, {200, 3779}, {213, 2333}, {373, 748}, {389, 3072}, {518, 3687}, {553, 1463}, {575, 5363}, {612, 3688}, {750, 3917}, {756, 2171}, {942, 5530}, {994, 1361}, {1124, 1685}, {1254, 1425}, {1317, 3032}, {1335, 1686}, {1356, 5213}, {1358, 3034}, {1364, 5348}, {1376, 4259}, {1395, 1843}, {1672, 1683}, {1673, 1684}, {1674, 1693}, {1675, 1694}, {1695, 1697}, {2007, 2019}, {2008, 2020}, {2330, 5285}, {2534, 2538}, {2535, 2539}, {3027, 3029}, {3028, 3031}, {3056, 5269}, {3340, 4517}, {3781, 5268}, {3792, 3819}, {4276, 5172}

▸ Functional Relations14 relations

• X₁₈₁ = isogonal conjugate of X₂₆₁

• X₁₈₁ = complement of X₃₅₆₁₄

• X₁₈₁ = X_i-Ceva conjugate of X_j for these (i,j): (12, 2197), (59, 4559), (65, 2171), (2171, 1500)

• X₁₈₁ = X_i-cross conjugate of X_j for these (i,j): (872, 1500), (2643, 512)

• X₁₈₁ = crosspoint of X_i and X_j for these (i,j): (42, 1824), (59, 4559), (65, 1400), (1354, 2171)

• X₁₈₁ = crosssum of X_i and X_j for these (i,j): (2, 2975), (11, 4560), (21, 333), (81, 4225), (86, 1444), (1098, 2185)

• X₁₈₁ = crossdifference of X₃₉₀₄ and X₃₉₁₀

• X₁₈₁ = X_i-beth conjugate of X_j for these (i,j): (42, 181), (660, 181), (756, 756)

• X₁₈₁ = {X₁,X₉₇₀}-harmonic conjugate of X₁₆₈₂

• X₁₈₁ = X₆₀-isoconjugate of X₇₅

• X₁₈₁ = trilinear product of vertices of extangents triangle

• X₁₈₁ = trilinear product of X_i and X_j for these {I,J}: {1, 181}, {6, 2171}, {7, 872}, {10, 1402}, {12, 31}, {19, 2197}, {25, 201}, {33, 1425}, {34, 3690}, {37, 1400}, {42, 65}, {55, 1254}, {56, 756}, {57, 1500}, {59, 2643}, {71, 1880}, {73, 1824}, {109, 4705}, {115, 2149}, {210, 1042}, {213, 226}, {225, 228}, {227, 2357}, {349, 2205}, {512, 4551}, {594, 604}, {608, 3949}, {651, 4079}, {661, 4559}, {762, 1412}, {798, 4552}, {1020, 3709}, {1089, 1397}, {1110, 1365}, {1214, 2333}, {1334, 1427}, {1395, 3695}, {1409, 1826}, {1415, 4024}, {1426, 2318}, {1441, 1918}, {3063, 4605}, {3124, 4564}, {4017, 4557}

• X₁₈₁ = barycentric product of X_i and X_j for these {I,J}: {1, 2171}, {2, 181}, {4, 2197}, {6, 12}, {7, 1500}, {9, 1254}, {10, 1400}, {19, 201}, {34, 3949}, {37, 65}, {42, 226}, {56, 594}, {57, 756}, {59, 115}, {71, 225}, {72, 1880}, {73, 1826}, {85, 872}, {109, 4024}, {210, 1427}, {213, 1441}, {227, 1903}, {278, 3690}, {281, 1425}, {307, 2333}, {321, 1402}, {349, 1918}, {512, 4552}, {523, 4559}, {604, 1089}, {608, 3695}, {651, 4705}, {661, 4551}, {663, 4605}, {664, 4079}, {762, 1014}, {1018, 4017}, {1020, 4041}, {1042, 2321}, {1091, 2150}, {1109, 2149}, {1214, 1824}, {1252, 1365}, {1262, 4092}, {1334, 3668}, {1404, 4013}, {1411, 4053}, {1415, 4036}, {1426, 3694}, {2222, 2610}, {2643, 4564}, {3124, 4998}, {3709, 4566}

• X₁₈₁ = X_i-isoconjugate of X_j for these (i,j): (1, 261), (2, 2185), (7, 1098), (8, 757), (9, 1509), (21, 86), (27, 1812), (28, 332), (29, 1444), (55, 873), (58, 314), (60, 75), (69, 270), (76, 2150), (81, 333), (99, 3737), (200, 552), (249, 4858), (274, 284), (283, 286), (304, 2189), (310, 2194), (312, 593), (348, 2326), (514, 4612), (645, 1019), (649, 4631), (650, 4610), (662, 4560), (663, 4623), (693, 4636), (763, 2321), (849, 3596), (1014, 1043), (1021, 4573), (1434, 2287), (2170, 4590), (4391, 4556)

Notes

X₁₈₁ is the external center of similitude (or exsimilicenter) of the incircle and Apollonius circle. The internal center is X₁₆₈₂. (Peter J. C. Moses, 8/22/2003)

X₁₈₁ is the isogonal conjugate of the isotomic conjugate of X₁₂; also, X₁₈₁ is the {X_i ,X_j }-harmonic conjugate of X_k for these (i,j,k)): (31,51,3271), (42,1400,1402), (57,1401,1357), (57,1469,1401). (Peter J. C. Moses, 6/20/2014)

A proof of the the concurrence of lines AA',BB',CC' follows.

A = exsimilicenter(incircle, A-excircle)

A' = exsimilicenter(A-excircle, Apollonius circle)

Let J = exsimilicenter(incircle, Apollonius circle).

By Monge’s theorem, the points A, A', J are collinear. In particular, J lies on line AA', and cyclically, J lies on lines BB' and CC'. Therefore, J = X₁₈₁. (Darij Grinberg, Hyacinthos, 7461, 8/10/03)

Let A' be the inverse-in-excircles-radical-circle of the midpoint of BC, and define B', C' cyclically. Triangle A'B'C' is inscribed in the Apollonius circle and homothetic to the intouch triangle at X₁₈₁. (Randy Hutson, March 21, 2019)

See also Clark Kimberling, Shiko Iwata, and Hidetosi Fukagawa, Problem 1091 and Solution, Crux Mathematicorum 13 (1987) 128-129; 217-218. [proposed 1985].