X₁₄₀₇

X₅₇-BETH CONJUGATE OF X₅₇

X₁₄₀₇ = isogonal conjugate of X₃₄₆

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

X₁₄₀₇ (X₅₇-BETH CONJUGATE OF X₅₇) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {6, 57}, {31, 56}

• Related Centers: X₅₇ (ISOGONAL CONJUGATE OF X₉), X₆ (SYMMEDIAN POINT), X₃₁ (2nd POWER POINT), X₅₆ (EXSIMILICENTER)

Trilinears: \((1 - \cos A)(1 - \cos A + \cos B + \cos C) : :\)

Search Value (6, 9, 13): \(0.025677058129\)

▸ Alternate Coordinates1 form

• Trilinears: \(a/(b + c - a)^{2} : :\)

▸ Incident Central Lines26 lines

{3, 951}, {6, 57}, {7, 940}, {31, 56}, {34, 1413}, {55, 1458}, {63, 220}, {73, 1466}, {81, 279}, {109, 1477}, {189, 1146}, {278, 1086}, {478, 1122}, {479, 1462}, {534, 553}, {608, 1435}, {614, 1456}, {739, 934}, {942, 1448}, {1104, 1394}, {1119, 1396}, {1333, 1412}, {1357, 1397}, {1398, 1408}, {1401, 1460}, {1464, 1470}

▸ Functional Relations11 relations

• X₁₄₀₇ = isogonal conjugate of X₃₄₆

• X₁₄₀₇ = complement of isotomic conjugate of X₃₄₅₄₆

• X₁₄₀₇ = X_i-Ceva conjugate of X_j for these (i,j): (269, 56), (1119, 1398), (1262, 1461), (1275, 934), (1396, 1435)

• X₁₄₀₇ = X_i-cross conjugate of X_j for these (i,j): (604, 56), (608, 1413), (1042, 269)

• X₁₄₀₇ = cevapoint of X₆₀₄ and X₁₁₀₆

• X₁₄₀₇ = crosspoint of X_i and X_j for these (i,j): (57, 1422), (269, 738), (279, 1119), (934, 1275), (1262, 1461), (1396, 1412)

• X₁₄₀₇ = crosssum of X_i and X_j for these (i,j): (200, 728), (220, 1260)

• X₁₄₀₇ = trilinear product X₅₆*X₅₇

• X₁₄₀₇ = X₉₂-isoconjugate of X₁₂₆₀

• X₁₄₀₇ = perspector of ABC and unary cofactor triangle of Ayme triangle

• X₁₄₀₇ = barycentric square of X₅₇

Notes

X₁₄₀₇ is the vertex conjugate of the foci of the inellipse that is the barycentric square of the Gergonne line (with center X₄₀₀₀ and perspector X₂₇₉). (Randy Hutson, October 15, 2018)