X₁₃₅₁

REFLECTION OF X₃ IN X₆

X₁₃₅₁ is the reflection of X₃ in X₆.

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

X₁₃₅₁ (REFLECTION OF X₃ IN X₆) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {3, 6}

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

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

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

Search Value (6, 9, 13): \(-4.796545904065\)

▸ Alternate Coordinates3 forms

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

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

• Barycentrics: \(a^{2}(S^{2} - S_{A}^{2} + S_{B} S_{C}) : :\)

▸ Linear Combinations1 combination

• X₁₃₅₁ = X₃ - \(2\) X₆

▸ Incident Central Lines11 lines

{3, 6}, {4, 193}, {5, 69}, {25, 110}, {30, 1353}, {49, 206}, {51, 394}, {159, 195}, {183, 262}, {381, 524}, {613, 999}

▸ Functional Relations9 relations

• X₁₃₅₁ = midpoint of X₄ and X₁₉₃

• X₁₃₅₁ = reflection of X_i in X_j for these (i,j): (3, 6), (6, 576), (69, 5), (1350, 182)

• X₁₃₅₁ = isogonal conjugate of X₇₆₁₂

• X₁₃₅₁ = inverse-in-2nd-Lemoine-circle of X₁₆₉₂

• X₁₃₅₁ = radical center of Lucas(-4 tan ω) circles

• X₁₃₅₁ = intersection of tangents to 2nd Lemoine circle at intersections with circumcircle

• X₁₃₅₁ = inverse-in-{circle centered at X₃ with radius |OK|} of X₁₈₂

• X₁₃₅₁ = {X_i,X_j}-harmonic conjugate of X_k for these (i,j,k): (6, 1350, 182), (1687, 1688, 5033)

• X₁₃₅₁ = exsimilicenter of circle centered at X₃₇₁ through X₁₁₅₂ and circle centered at X₃₇₂ through X₁₁₅₁; the insimilicenter is X₃₀₅₃

Notes

Let T be a triangle inscribed in the circumcircle and circumscribing the orthic inconic. As T varies, its orthocenter traces a circle centered at X₁₃₅₁ with segment X₄X₁₉₃ as diameter. (Randy Hutson, August 29, 2018)