X₃₆₆₃

X₆com(INTOUCH TRIANGLE)

X₃₆₆₃ = trilinear product of vertices of inverse-in-excircles triangle

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

X₃₆₆₃ (X₆com(INTOUCH TRIANGLE)) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Circle: Incircle (incircle) — Inscribed circle tangent to sidelines at T_a, T_b, T_c; centered at X₁ INCENTER

• Central Lines: {1, 7}, {10, 75}

• Related Centers: X₁ (INCENTER), X₁₂₂₂ (ISOGONAL CONJUGATE OF X₁₂₀₁), X₇ (GERGONNE POINT), X₁₀ (SPIEKER CENTER), X₇₅ (ISOTOMIC CONJUGATE OF INCENTER)

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

Search Value (6, 9, 13): \(2.040978573191\)

▸ Incident Central Lines28 lines

{1, 7}, {2, 2415}, {6, 527}, {9, 3008}, {10, 75}, {37, 142}, {57, 1766}, {63, 1723}, {69, 519}, {86, 99}, {141, 536}, {144, 1743}, {165, 3598}, {226, 1465}, {256, 2481}, {273, 1785}, {307, 1210}, {319, 3625}, {320, 3244}, {329, 2999}, {545, 3589}, {553, 940}, {572, 1429}, {573, 1423}, {988, 1125}, {1074, 1111}, {1122, 3057}, {1439, 2823}

▸ Functional Relations6 relations

• X₃₆₆₃ = isotomic conjugate of X₁₂₂₂

• X₃₆₆₃ = anticomplement of X₁₇₃₅₅

• X₃₆₆₃ = trilinear product of vertices of inverse-in-excircles triangle

• X₃₆₆₃ = perspector of ABC and cross-triangle of ABC and inverse-in-excircles triangle

• X₃₆₆₃ = complement of X₃₇₂₉

• X₃₆₆₃ = complement, wrt intouch triangle, of X₁₂₇₂₃

Notes

Let A'B'C' be the inverse-in-excircles triangle. Let A″ be the trilinear product B'*C', and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X₃₆₆₃. (Randy Hutson, July 20, 2016)