X₁₃₅₀

REFLECTION OF X₆ IN X₃

X₁₃₅₀ is the reflection of X₆ in X₃.

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

X₁₃₅₀ (REFLECTION OF X₆ IN X₃) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {6, 3}

• Related Centers: X₆ (SYMMEDIAN POINT), X₃ (CIRCUMCENTER)

Trilinears: \(a - 4R \cot \omega \cos A : :\)

Barycentrics: \(a^{2}(a^{4} - 3b^{4} - 3c^{4} + 2a^{2}b^{2} + 2a^{2}c^{2} - 2b^{2}c^{2}) : :\)

Search Value (6, 9, 13): \(12.571817293327\)

▸ Alternate Coordinates2 forms

• Trilinears: \(\sin A - 2 \cos A \cot \omega : :\)

• Trilinears: \(h(A,B,C) : h(B,C,A) : h(C,A,B)\), where \(h(A,B,C) = 2 \cos A - \sin A \tan \omega\) (Peter J. C. Moses, 8/22/03)

▸ Linear Combinations1 combination

• X₁₃₅₀ = \(2\) X₃ - X₆

▸ Incident Central Lines15 lines

{2, 3066}, {3, 6}, {4, 141}, {20, 64}, {22, 110}, {30, 599}, {35, 611}, {36, 613}, {40, 518}, {74, 1296}, {103, 1293}, {206, 1092}, {343, 1370}, {376, 524}, {517, 990}

▸ Functional Relations10 relations

• X₁₃₅₀ = midpoint of X₂₀ and X₆₉

• X₁₃₅₀ = reflection of X_i in X_j for these (i,j): (4, 141), (6, 3), (1351, 182), (1498, 159)

• X₁₃₅₀ = isogonal conjugate of X₃₄₂₄

• X₁₃₅₀ = X₆ of circumcevian triangle of X₅₁₁

• X₁₃₅₀ = radical center of Lucas(-tan ω) circles

• X₁₃₅₀ = {X₁₈₂,X₁₃₅₁}-harmonic conjugate of X₆

• X₁₃₅₀ = inverse-in-2nd-Brocard-circle of X₅₁₈₈

• X₁₃₅₀ = antipedal-isogonal conjugate of X₆

• X₁₃₅₀ = X₅₃-of-the-hexyl-triangle

• X₁₃₅₀ = exsimilicenter of circle centered at X₁₁₅₁ through X₃₇₂ and circle centered at X₁₁₅₂ through X₃₇₁; the insimilicenter is X₃₀₅₃

Notes

Let A'B'C' be the reflection of ABC in X₃. Let A_B = BC∩C'A', and define B_C and C_A cyclically. A_C = BC∩A'B', and define B_A and C_B cyclically. The 6 points A_B, B_C, C_A, A_C, B_A, C_B lie on a conic. Let A″ be the intersection of the tangents to the conic at A_B and A_C, and define B″, C″ cyclically. The lines A'A″, B'B″, C'C″ concur in X₁₃₅₀. (Randy Hutson, January 29, 2015)