X₁₆₅₅

2nd HATZIPOLAKIS PARALLELIAN POINT

Continuing from the description of X₁₆₅₄, let h(B,A) be the distance from the point B_A to the line CA, and define five other distances cyclically. If P = X₁₆₅₅, then

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

X₁₆₅₅ (2nd HATZIPOLAKIS PARALLELIAN POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Derived Triangle: Anticomplementary Triangle △K_aK_bK_c — vertices K_a, K_b, K_c (Formed by lines through A, B, C parallel to opposite sides)

• Central Lines: {2, 274}

• Related Centers: X₂ (CENTROID), X₂₇₄ (ISOGONAL CONJUGATE OF X₂₁₃)

Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\), where \(f(a,b,c) = bc[-1/a^{2} + 1/b^{2} + 1/c^{2} + 1/(bc) + 1/(ca) + 1/(ab)]\)

Barycentrics: \(g(a,b,c) : g(b,c,a) : g(c,a,b)\), where \(g(a,b,c) = -1/a^{2} + 1/b^{2} + 1/c^{2} + 1/(bc) + 1/(ca) + 1/(ab)\)

Search Value (6, 9, 13): \(1.445819564171\)

▸ Incident Central Lines6 lines

{2, 39}, {8, 192}, {21, 385}, {193, 452}, {350, 1107}, {668, 1500}

▸ Functional Relations4 relations

• X₁₆₅₅ = anticomplement of X₂₇₄

• X₁₆₅₅ = isotomic conjugate of isogonal conjugate of X₂₁₇₇₉

• X₁₆₅₅ = polar conjugate of isogonal conjugate of X₂₃₀₇₉

• X₁₆₅₅ = X_i-Ceva conjugate of X_j for these (i,j): (37, 2), (1909, 8)

Notes

h(B,A) + \(h(C,A) = h(C\),B) + \(h(A,B) = h(A\),C) + h(B,C) (Antreas Hatzipolakis, Anopolis #20, 1/20/02)