X₁₄₃₉

X₃₀₇-BETH CONJUGATE OF X₃₀₇

X₁₄₃₉ = crosspoint of X₇ and X₇₇

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

X₁₄₃₉ (X₃₀₇-BETH CONJUGATE OF X₃₀₇) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {4, 7}, {6, 57}

• Related Centers: X₃₀₇ (ISOTOMIC CONJUGATE OF X₂₉), X₄ (ORTHOCENTER), X₇ (GERGONNE POINT), X₆ (SYMMEDIAN POINT), X₅₇ (ISOGONAL CONJUGATE OF X₉)

Trilinears: \((1 - \cos A)u(a,b,c) : :\), where \(u : v : w\) = X₃₀₇

Search Value (6, 9, 13): \(-0.393381924267\)

▸ Alternate Coordinates1 form

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

▸ Incident Central Lines16 lines

{1, 64}, {3, 77}, {4, 7}, {6, 57}, {37, 1020}, {54, 1443}, {71, 1214}, {72, 307}, {74, 934}, {86, 658}, {241, 579}, {284, 1461}, {347, 517}, {1014, 1175}, {1042, 1245}, {1088, 1246}

▸ Functional Relations8 relations

• X₁₄₃₉ = X_i-Ceva conjugate of X_j for these (i,j): (658, 905), (1446, 1427)

• X₁₄₃₉ = X_i-cross conjugate of X_j for these (i,j): (73, 1214), (656, 1020), (1410, 1427)

• X₁₄₃₉ = crosspoint of X₇ and X₇₇

• X₁₄₃₉ = crosssum of X_i and X_j for these (i,j): (24, 204), (33, 55)

• X₁₄₃₉ = isogonal conjugate of X₄₁₈₃

• X₁₄₃₉ = perspector of intouch triangle and 3rd extouch triangle

• X₁₄₃₉ = perspector of ABC and cross-triangle of ABC and 3rd extouch triangle

• X₁₄₃₉ = trilinear product of Jerabek hyperbola intercepts of Soddy line

Notes

Let A' be the homothetic center of the orthic triangles of the intouch and A-extouch triangles, and define B' and C' cyclically. The triangle A'B'C' is perspective to the intouch triangle at X₁₄₃₉. (Randy Hutson, September 14, 2016)