X₃₁₄

ISOTOMIC CONJUGATE OF X₆₅

X₃₁₄ is the isotomic conjugate of X₆₅.

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

X₃₁₄ (ISOTOMIC CONJUGATE OF X₆₅) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {4, 69}, {1, 75}

• Related Centers: X₆₅ (ORTHOCENTER OF THE INTOUCH TRIANGLE), X₄ (ORTHOCENTER), X₆₉ (SYMMEDIAN POINT OF THE ANTICOMPLEMENTARY TRIANGLE), X₁ (INCENTER), X₇₅ (ISOTOMIC CONJUGATE OF INCENTER)

Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\), where \(f(a,b,c) = b^{2}c^{2}(b + c - a)/(b + c)\)

Barycentrics: \(g(a,b,c) : g(b,c,a) : g(c,a,b)\), where \(g(a,b,c) = bc(b + c - a)/(b + c)\)

Search Value (6, 9, 13): \(5.033660717966\)

▸ Incident Central Lines16 lines

{1, 75}, {2, 941}, {4, 69}, {6, 981}, {7, 310}, {9, 312}, {21, 261}, {29, 1039}, {58, 987}, {79, 320}, {80, 313}, {81, 321}, {84, 309}, {99, 104}, {256, 350}, {294, 645}

▸ Functional Relations14 relations

• X₃₁₄ = isogonal conjugate of X₁₄₀₂

• X₃₁₄ = isotomic conjugate of X₆₅

• X₃₁₄ = anticomplement of X₂₀₉₂

• X₃₁₄ = X₃₁₀-Ceva conjugate of X₂₇₄

• X₃₁₄ = cevapoint of X_i and X_j for these (i,j): (8, 312), (69, 75)

• X₃₁₄ = X_i-cross conjugate of X_j for these (i,j): (8, 333), (69, 332), (333, 274), (497, 29)

• X₃₁₄ = crosspoint of X₁ and X₁₇₆₄ wrt excentral triangle

• X₃₁₄ = crosspoint of X₁ and X₁₇₆₄ wrt anticomplementary triangle

• X₃₁₄ = pole wrt polar circle of trilinear polar of X₁₈₈₀

• X₃₁₄ = X₄₈-isoconjugate (polar conjugate) of X₁₈₈₀

• X₃₁₄ = perspector of 4th Conway triangle and cross-triangle of ABC and 4th Conway triangle

• X₃₁₄ = perspector of 5th Conway triangle and cross-triangle of ABC and 5th Conway triangle

• X₃₁₄ = perspector of inverse-in-Conway-circle triangle and cross-triangle of ABC and inverse-in-Conway-circle triangle

• X₃₁₄ = barycentric product X₉₉*X₄₃₉₁

Notes

Let D and E be the intersections of line X₁X₃ with lines PU(3) and PU(6), respectively. Let D' and E' be the isogonal conjugates of D and E, respectively. Let D″ and E″ be the isotomic conjugates of D and E, respectively. Then X₃₁₄ = D'D″ ∩E'E″. (Randy Hutson, December 26, 2015)

Let A₄B₄C₄ be the 4th Conway triangle. Let A' be the cevapoint of B₄ and C₄, and define B' and C' cyclically. The lines AA', BB', CC' concur in X₃₁₄. (Randy Hutson, December 10, 2016)

Let A₅B₅C₅ be the 5th Conway triangle. Let A' be the cevapoint of B₅ and C₅, and define B' and C' cyclically. The lines AA', BB', CC' concur in X₃₁₄. (Randy Hutson, December 10, 2016)