X₁₀₇₁

INTERSECTION OF LINES X₁X_84 AND X₄X_7

X₁₀₇₁ = crosspoint of X₇ and X₆₃

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

X₁₀₇₁ (INTERSECTION OF LINES X₁X_84 AND X₄X_7) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {4, 7}, {6, 63}

• Related Centers: X₄ (ORTHOCENTER), X₇ (GERGONNE POINT), X₆ (SYMMEDIAN POINT), X₆₃ (ISOGONAL CONJUGATE OF X₁₉)

Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\), where \(f(a,b,c) = (b^{2} + c^{2} - a^{2})[a^{3}(b + c) - (b - c)^{2}(a^{2} + a(b + c) - (b + c)^{2})][a^{3} - (b + c)(a^{2} + a(b + c) - (b - c)^{2})]\)

Barycentrics: \(af(a,b,c) : bf(b,c,a) : cf(c,a,b)\)

Search Value (6, 9, 13): \(7.896654233889\)

▸ Incident Central Lines16 lines

{1, 84}, {4, 7}, {6, 63}, {10, 2801}, {20, 145}, {21, 104}, {27, 1871}, {198, 1741}, {227, 1735}, {355, 377}, {412, 1872}, {496, 1519}, {774, 1458}, {910, 1729}, {1210, 1532}, {1317, 1364}

▸ Functional Relations8 relations

• X₁₀₇₁ = reflection of X_i in X_j for these (i,j): (4, 942), (72, 3)

• X₁₀₇₁ = isotomic conjugate of isogonal conjugate of X₂₃₂₀₄

• X₁₀₇₁ = crosspoint of X₇ and X₆₃

• X₁₀₇₁ = crosssum of X_i and X_j for these (i,j): (1, 1777), (19, 55), (25, 2331)

• X₁₀₇₁ = X₆₈-of-intouch triangle

• X₁₀₇₁ = X₂₀-of-X₁-Brocard triangle

• X₁₀₇₁ = intouch isogonal conjugate of X₅₄

• X₁₀₇₁ = intouch isotomic conjugate of X₁₂₇₂₃

Notes

X₁₀₇₁ appears in Hyacinthos message #3849, Paul Yiu, Sept. 19, 2001.