X₃₆₁₇

H(X₁; M, -2)

X₃₆₁₇ = insimilicenter of Spieker circle and AC-incircle

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

X₃₆₁₇ (H(X₁; M, -2)) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {1, 2}, {20, 355}

• Related Centers: X₁ (INCENTER), X₂ (CENTROID), X₂₀ (DE LONGCHAMPS POINT), X₃₅₅ (FUHRMANN CENTER)

Trilinears: \(bc(a - 3b - 3c) : :\)

Barycentrics: \(a - 3b - 3c : b - 3c - 3a : c - 3a - 3b\)

Search Value (6, 9, 13): \(3.380617018914\)

▸ Alternate Coordinates1 form

• Trilinears: \(2 r - 3 R \sin B \sin C : :\)

▸ Linear Combinations1 combination

• X₃₆₁₇ = \(4\) X₁ - \(9\) X₂ = X₇ + \(2\) X₈ + \(2\) X₉ = X₈ + \(4\) X₁₀

▸ Incident Central Lines29 lines

{1, 2}, {20, 355}, {40, 3146}, {44, 391}, {45, 346}, {63, 1706}, {88, 1219}, {100, 958}, {144, 1654}, {149, 1145}, {193, 3416}, {321, 341}, {377, 3421}, {390, 1837}, {404, 956}, {442, 1159}, {452, 3419}, {496, 1000}, {515, 3522}, {517, 3091}, {631, 952}, {944, 3523}, {984, 1278}, {1376, 2975}, {1482, 3090}, {1483, 3526}, {1697, 3305}, {1788, 3600}, {2551, 3434}

▸ Functional Relations3 relations

• X₃₆₁₇ = isotomic conjugate of X₃₀₇₁₂

• X₃₆₁₇ = insimilicenter of Spieker circle and AC-incircle

• X₃₆₁₇ = {X_i,X_j}-harmonic conjugate of X_k for these (i,j,k): (2, 8, 145), (8, 10, 2)