X₂₅₅₀

INSIMILICENTER(SPIEKER CIRCLE, (X₄,2R))

The circle (X₄,2R) is identified at X₂₄₇₄.

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

X₂₅₅₀ (INSIMILICENTER(SPIEKER CIRCLE, (X₄,2R))) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {7, 8}

• Related Centers: X₇ (GERGONNE POINT), X₈ (NAGEL POINT)

Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\), where \(f(a,b,c) = bc(b + c + a \cos B \cos C)\)

Barycentrics: \(af(a,b,c) : bf(b,c,a) : cf(c,a,b)\)

Search Value (6, 9, 13): \(2.625825168779\)

▸ Incident Central Lines35 lines

{1, 142}, {2, 11}, {3, 1602}, {4, 9}, {7, 8}, {12, 480}, {20, 958}, {144, 1654}, {200, 226}, {210, 329}, {318, 1118}, {355, 971}, {376, 993}, {442, 954}, {519, 1056}, {527, 1478}, {740, 2294}, {936, 946}, {948, 2263}, {960, 962}, {1058, 1125}, {1377, 1588}, {1378, 1587}, {1445, 1788}, {1479, 1698}, {1573, 2549}, {1574, 2548}, {1678, 2543}, {1679, 2542}, {1680, 2547}, {1681, 2546}, {2013, 2545}, {2014, 2544}, {2467, 2553}, {2468, 2552}

▸ Functional Relations7 relations

• X₂₅₅₀ = midpoint of X₇ and X₈

• X₂₅₅₀ = reflection of X_i in X_j for these (i,j): (1, 142), (9, 10), (390, 1001)

• X₂₅₅₀ = isogonal conjugate of X₃₄₂₃

• X₂₅₅₀ = complement of X₃₉₀

• X₂₅₅₀ = anticomplement of X₁₀₀₁

• X₂₅₅₀ = inverse-in-Feuerbach-hyperbola of X₄₉₇

• X₂₅₅₀ = outer-Garcia-to-ABC similarity image of X₉

Notes

Let A'B'C' be the intangents-to-extangents similarity image of ABC. A'B'C' is homothetic to ABC at X₅₅ and to the medial triangle at X₂₅₅₀. (Randy Hutson, December 2, 2017)

Let A'B'C' be the intouch triangle of the extangents triangle, if ABC is acute. Then A'B'C' is homothetic to the anticomplementary triangle at X₂₅₅₀. (Randy Hutson, December 2, 2017)