X₆₂₇

ANTICOMPLEMENT OF X₁₇

X₆₂₇ is the anticomplement of X₁₇.

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

X₆₂₇ (ANTICOMPLEMENT OF X₁₇) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {17, 629}

• Related Centers: X₁₇ (1st NAPOLEON POINT), X₆₂₉ (COMPLEMENT OF X₁₇)

Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\), where \(f(a,b,c) = [S_{B}S_{C} + 2S_{A}(a^{2} + \sqrt{3} \operatorname{area})]/a\)

Barycentrics: \(g(a,b,c) : g(b,c,a) : g(c,a,b)\), where \(g(a,b,c) = af(a,b,c)\)

Search Value (6, 9, 13): \(5.114797277327\)

▸ Alternate Coordinates1 form

• Trilinears: \(F(17)/a - 2 \sec(A - \pi /3) : F(17)/b - 2 \sec(B - \pi /3) : F(17)/c - 2 \sec(C - \pi /3)\)

X₆₂₇ lies on the Napoleon cubic.

▸ Incident Central Lines9 lines

{2, 17}, {3, 298}, {4, 616}, {5, 302}, {16, 635}, {20, 621}, {54, 69}, {61, 618}, {140, 299}

▸ Functional Relations7 relations

• X₆₂₇ = reflection of X₁₇ in X₆₂₉

• X₆₂₇ = isogonal conjugate of X₃₄₈₉

• X₆₂₇ = anticomplement of X₁₇

• X₆₂₇ = anticomplementary conjugate of X₆₃₃

• X₆₂₇ = X_i-Ceva conjugate of X_j for these (i,j): (5, 628), (302, 2)

• X₆₂₇ = {X₅₄,X₁₂₇₃}-harmonic conjugate of X₆₂₈

• X₆₂₇ = {X₆₉,X₆₃₁}-harmonic conjugate of X₆₂₈

Notes

Erect equilateral triangles outwards on the sides of triangle ABC; the circumcenter of the apices is X₆₂₇. (Peter Moses, 7/16,2003)

Let A' be the isogonal conjugate of the A-vertex of the inner Napoleon triangle, and define B' and C' cyclically. Triangle A'B'C' is perspective to the inner Napoleon triangle at X₅, and to the outer Napoleon triangle at X₆₂₇. (Randy Hutson, June 7, 2019)

Let O_A be the circle centered at the A-vertex of the outer Fermat triangle and passing through A; define O_B and O_C cyclically. X₆₂₇ is the radical center of O_A, O_B, O_C. (Randy Hutson, August 30, 2020)