X₁₆₈₂

INSIMILICENTER(INCIRCLE, APOLLONIUS CIRCLE)

The exsimilicenter of the incircle and Apollonius circle is X₁₈₁. Also, the triangle A'B'C' formed (as at X₂₀₉₂ by the intersections of the Apollonius circle and the excircles is perspective to the cevian triangle of X₁, and the perspector is X₁₆₈₂. (Paul Yiu, Hyacinthos #8076, 10/01/03)

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

X₁₆₈₂ (INSIMILICENTER(INCIRCLE, APOLLONIUS CIRCLE)) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Derived Triangle: Cevian Triangle △P_aP_bP_c — vertices P_a, P_b, P_c (Traces of cevians through target center on BC, CA, AB)

• Concurrence Lines: A–P_a, B–P_b, C–P_c concurring at X₁₆₈₂

Trilinears: \([s \cos(A/2) - r \sin(A/2)]^{2} : :\)

Search Value (6, 9, 13): \(2.864369413485\)

▸ Alternate Coordinates1 form

• Trilinears: \(a(b + c - a)(b^{2} + c^{2} + ab + ac)^{2} : :\) (M. Iliev, 5/13/07)

▸ Incident Central Lines19 lines

{1, 181}, {3, 1397}, {10, 11}, {43, 1697}, {55, 386}, {56, 573}, {57, 1695}, {73, 1362}, {212, 1472}, {215, 501}, {988, 1401}, {1124, 1686}, {1335, 1685}, {1672, 1684}, {1673, 1683}, {1674, 1694}, {1675, 1693}, {2007, 2020}, {2008, 2019}

▸ Functional Relations3 relations

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

• X₁₆₈₂ = perspector of ABC and cross-triangle of ABC and Apollonius triangle

• X₁₆₈₂ = centroid of curvatures of Apollonius circle and excircles

Notes

Let J_aJ_bJ_c be the excentral triangle and P_aP_bP_c be the Apollonius triangle. Let P_a' = {X₉₇₀,J_a}-harmonic conjugate of P_a, and define P_b' and P_c' cyclically. The lines AP_a', BP_b', CP_c' concur in X₁₆₈₂; see also X₁₁. (Randy Hutson, December 10, 2016)