X₂₇₅

CEVAPOINT OF ORTHOCENTER AND SYMMEDIAN POINT

X₂₇₅ is the cevapoint of orthocenter and symmedian point.

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

X₂₇₅ (CEVAPOINT OF ORTHOCENTER AND SYMMEDIAN POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {4, 54}, {2, 95}

• Related Centers: X₄ (ORTHOCENTER), X₆ (SYMMEDIAN POINT), X₅₄ (KOSNITA POINT), X₂ (CENTROID), X₉₅ (CEVAPOINT OF CENTROID AND CIRCUMCENTER)

Trilinears: \(\csc 2A \sec(B - C) : \csc 2B \sec(C - A) : \csc 2C \sec(A - B)\)

Barycentrics: \(\sec A \sec(B - C) : \sec B \sec(C - A) : \sec C \sec(A - B)\)

Search Value (6, 9, 13): \(-6.110630103881\)

▸ Alternate Coordinates2 forms

• Trilinears: \(\sec(A + T) : :\), T as at X₃₈₉

• Barycentrics: \(1/((a^{2} - b^{2} - c^{2}) (a^{2} (b^{2} + c^{2}) - (b^{2} - c^{2})^{2})) : :\)

▸ Incident Central Lines13 lines

{2, 95}, {4, 54}, {13, 472}, {14, 473}, {17, 471}, {18, 470}, {25, 262}, {51, 107}, {53, 288}, {76, 276}, {83, 297}, {94, 324}, {98, 427}

▸ Functional Relations13 relations

• X₂₇₅ = isogonal conjugate of X₂₁₆

• X₂₇₅ = isotomic conjugate of X₃₄₃

• X₂₇₅ = X₂₇₆-Ceva conjugate of X₉₅

• X₂₇₅ = cevapoint of X₄ and X₆

• X₂₇₅ = X_i-cross conjugate of X_j for these (i,j): (6, 54), (54, 95)

• X₂₇₅ = crosssum of X₂₁₇ and X₄₁₈

• X₂₇₅ = trilinear pole of line X₁₈₆X₅₂₃ (the polar of X₄ wrt the circumcircle, and the polar of X₅ wrt polar circle)

• X₂₇₅ = crosspoint wrt tangential triangle of X₄ and X₆

• X₂₇₅ = intersection of tangents at X₄ and X₆ to bianticevian conic of X₄ and X₆

• X₂₇₅ = pole wrt polar circle of trilinear polar of X₅ (line X₂₀₈₁X₂₆₀₀)

• X₂₇₅ = polar conjugate of X₅

• X₂₇₅ = X₁₆₃-isoconjugate of X₆₃₆₈

• X₂₇₅ = barycentric product of circumcircle intercepts of line X₃₄₀X₅₂₀

Notes

Let A'B'C' be the circumorthic triangle. Let A″ be the trilinear pole of line B'C', and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X₂₇₅. (Randy Hutson, June 7, 2019)