X₁₆₅₄

1st HATZIPOLAKIS PARALLELIAN POINT

Let P be a point in the plane of, but not on a sideline of, triangle ABC. Let B_A be the point where the line through P parallel to line BC meets line BA, and let C_A be the point where the line through P parallel to line BC meets line CA. Define C_B, A_B, A_C, and B_C cyclically. If P = X₁₆₅₄, then

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

X₁₆₅₄ (1st HATZIPOLAKIS PARALLELIAN POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {86, 1213}

• Related Centers: X₈₆ (CEVAPOINT OF INCENTER AND CENTROID), X₁₂₁₃ (ISOGONAL CONJUGATE OF X₁₁₇₁)

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

Barycentrics: \(g(a,b,c) : g(b,c,a) : g(c,a,b)\), where \(g(a,b,c) = -a^{2} + b^{2} + c^{2} + bc + ca + ab\)

Search Value (6, 9, 13): \(3.535521057858\)

▸ Incident Central Lines6 lines

{2, 6}, {8, 192}, {10, 894}, {37, 319}, {71, 1762}, {190, 594}

▸ Functional Relations11 relations

• X₁₆₅₄ = reflection of X₈₆ in X₁₂₁₃

• X₁₆₅₄ = isotomic conjugate of X₆₆₂₅

• X₁₆₅₄ = crosspoint of X₂ and X₈ wrt 2nd Conway triangle

• X₁₆₅₄ = anticomplement of X₈₆

• X₁₆₅₄ = X_i-Ceva conjugate of X_j for these (i,j): (10, 2), (894, 192)

• X₁₆₅₄ = anticomplementary isotomic conjugate of X₁

• X₁₆₅₄ = complement of X₂₀₀₉₀

• X₁₆₅₄ = polar conjugate of isogonal conjugate of X₂₂₁₃₉

• X₁₆₅₄ = {X₂,X₂₈₉₅}-harmonic conjugate of X₁₇₇₇₈

• X₁₆₅₄ = perspector of Gemini triangle 39 and cross-triangle of Gemini triangles 39 and 40

• X₁₆₅₄ = {X₂,X₆₉}-harmonic conjugate of X₁₇₃₀₀

Notes

|AB_A| + |AC_A| = |BC_B| + |BA_B| = |CA_C| + |CB_C| (Antreas Hatzipolakis, Anopolis #20, 1/20/02)