X₃₉₄₅

X₃₂₄₇com(INTOUCH TRIANGLE)

X₃₉₄₅ = intersection of tangents at X₂ and X₇ to Lucas cubic K007

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

X₃₉₄₅ (X₃₂₄₇com(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}, {2, 6}

• Related Centers: X₁ (INCENTER), X₇ (GERGONNE POINT), X₂ (CENTROID), X₆ (SYMMEDIAN POINT)

Barycentrics: \(b^{2} + c^{2} - 3a^{2} - 2ab - 2ac - 2bc : :\)

Search Value (6, 9, 13): \(1.577621275493\)

▸ Incident Central Lines23 lines

{1, 7}, {2, 6}, {3, 1014}, {37, 144}, {57, 2269}, {75, 145}, {142, 1449}, {171, 1253}, {226, 1419}, {319, 3617}, {320, 1279}, {344, 3758}, {346, 894}, {354, 3056}, {522, 4724}, {527, 3247}, {941, 980}, {1002, 3779}, {1418, 3666}, {1433, 1440}, {1434, 3522}, {1456, 3485}, {3475, 3745}

▸ Functional Relations4 relations

• X₃₉₄₅ = intersection of tangents at X₂ and X₇ to Lucas cubic K007

• X₃₉₃₅ = inverse-in-circumconic-centered-at-X₁ of X₂

• X₃₉₄₅ = {X₂,X₆}-harmonic conjugate of X₃₇₆₈₁

• X₃₉₃₅ = endo-homothetic center of X₂- and X₃-Ehrmann triangles; the homothetic center is X₁₅₁₃₉