X₆₂₈

ANTICOMPLEMENT OF X₁₈

X₆₂₈ is the anticomplement of X₁₈.

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

X₆₂₈ (ANTICOMPLEMENT OF X₁₈) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {18, 630}

• Related Centers: X₁₈ (2nd NAPOLEON POINT), X₆₃₀ (COMPLEMENT OF X₁₈)

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): \(40.449454373023\)

▸ Alternate Coordinates1 form

• Trilinears: \(F(18)/a + 2 \sec(A + \pi /3) : F(18)/b + 2 \sec(B + \pi /3) : F(18)/c + 2 \sec(C + \pi /3)\)

X₆₂₈ lies on the Napoleon cubic.

▸ Incident Central Lines9 lines

{2, 18}, {3, 299}, {4, 617}, {5, 303}, {15, 636}, {20, 622}, {54, 69}, {62, 619}, {140, 298}

▸ Functional Relations7 relations

• X₆₂₈ = reflection of X₁₈ in X₆₃₀

• X₆₂₈ = isogonal conjugate of X₃₄₉₀

• X₆₂₈ = anticomplement of X₁₈

• X₆₂₈ = anticomplementary conjugate of X₆₃₄

• X₆₂₈ = X_i-Ceva conjugate of X_j for these (i,j): (5, 627), (303, 2)

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

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

Notes

Erect equilateral triangles inwards on the sides of triangle ABC; the circumcenter of the apices is X₆₂₈. (Peter Moses, 7/16,2003)

Let A' be the isogonal conjugate of the A-vertex of the outer Napoleon triangle, and define B' and C' cyclically. Triangle A'B'C' is perspective to the outer Napoleon triangle at X₅, and to the inner Napoleon triangle at X₆₂₈. (Randy Hutson, June 7, 2019)

Let O_A be the circle centered at the A-vertex of the inner Fermat triangle and passing through A; define O_B and O_C cyclically. X₆₂₈ is the radical center of O_A, O_B, O_C. (Randy Hutson, August 30, 2020)