X₁₄₆

REFLECTION OF X₂₀ IN X₁₁₀

X₁₄₆ is the {X₇₄,X₁₁₃}-harmonic conjugate of X₂. For a list of harmonic conjugates, click Tables at the top of this page.

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

X₁₄₆ (REFLECTION OF X₂₀ IN X₁₁₀) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {20, 110}

• Related Centers: X₂₀ (DE LONGCHAMPS POINT), X₁₁₀ (FOCUS OF KIEPERT PARABOLA)

Trilinears: \(bc(-avw + bwu + cuv) : :\), where \(u = u(A,B,C) = \cos A - 2 \cos B \cos C\), \(v = u(B,C,A)\), \(w = u(C,A,B)\)

Barycentrics: \(-avw + bwu + cuv : -bwu + cuv + avw : -cuv + avw + bwu\)

Search Value (6, 9, 13): \(-17.426219251116\)

▸ Alternate Coordinates1 form

• Barycentrics: \(a^{10} + a^{8} (b^{2} + c^{2}) - a^{6} (8 b^{4} - 9 b^{2} c^{2} + 8 c^{4}) + 2 a^{4} (b^{2} + c^{2}) (4 b^{4} - 7 b^{2} c^{2} + 4 c^{4}) - a^{2} (b^{2} - c^{2})^{2} (b^{4} + 9 b^{2} c^{2} + c^{4}) - (b^{2} - c^{2})^{4} (b^{2} + c^{2}) : :\)

X₁₄₆ lies on the anticomplementary circle.

▸ Incident Central Lines6 lines

{2, 74}, {4, 94}, {20, 110}, {30, 323}, {147, 690}, {148, 193}

▸ Functional Relations6 relations

• X₁₄₆ = reflection of X_i in X_j for these (i,j): (20, 110), (74, 113), (265, 1539)

• X₁₄₆ = isogonal conjugate of X₃₄₁₇₈

• X₁₄₆ = isotomic conjugate of anticomplement of X₃₆₈₉₆

• X₁₄₆ = anticomplementary conjugate of X₃₀

• X₁₄₆ = X₇₄-of-anticomplementary triangle

• X₁₄₆ = crosspoint of X₃₉₉ and X₂₉₃₅ wrt both the excentral and tangential triangles

Notes

Let T be the P-Brocard triangle for any P on the Euler line. Let O_A be the circle centered at the A-vertex of T and passing through A; define O_B and O_C cyclically. X₁₄₆ is the radical center of O_A, O_B, O_C. (Randy Hutson, August 30, 2020)