X₂₃₅

X₄-CEVA CONJUGATE OF X₁₈₅

X₂₃₅ = midpoint of X₄ and X₂₄

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

X₂₃₅ (X₄-CEVA CONJUGATE OF X₁₈₅) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {4, 24}, {2, 3} (Euler line)

• Related Centers: X₄ (ORTHOCENTER), X₂₄ (PERSPECTOR OF ABC AND ORTHIC-OF-ORTHIC TRIANGLE), X₂ (CENTROID), X₃ (CIRCUMCENTER)

Trilinears: \(\sec A - \cos(B - C) : :\)

Barycentrics: \((\tan A)(\cos^{2} B + \cos^{2} C) : :\)

Search Value (6, 9, 13): \(-2.042726748267\)

▸ Alternate Coordinates2 forms

• Trilinears: \((\sec A)(\cos^{2} B + \cos^{2} C) : :\)

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

▸ Incident Central Lines5 lines

{2, 3}, {11, 34}, {12, 33}, {52, 113}, {133, 136}

▸ Functional Relations16 relations

• X₂₃₅ = midpoint of X₄ and X₂₄

• X₂₃₅ = complement of X₁₁₄₁₃

• X₂₃₅ = anticomplement of X₁₆₁₉₆

• X₂₃₅ = circumcircle-inverse of X₃₇₉₁₇

• X₂₃₅ = excentral-to-ABC functional image of X₅₅

• X₂₃₅ = X₄-Ceva conjugate of X₁₈₅

• X₂₃₅ = crosssum of X₃ and X₁₀₉₂

• X₂₃₅ = orthic-isogonal conjugate of X₁₈₅

• X₂₃₅ = X₅₆ of orthic triangle if ABC is acute

• X₂₃₅ = insimilicenter of nine-point circle and incircle of orthic triangle if ABC is acute; the exsimilicenter is X₄₂₇

• X₂₃₅ = pole wrt polar circle of trilinear polar of X₈₀₁ (line X₅₂₃X₂₀₇₁)

• X₂₃₅ = X₄₈-isoconjugate (polar conjugate) of X₈₀₁

• X₂₃₅ = perspector of ABC and cross-triangle of ABC and 2nd Hyacinth triangle

• X₂₃₅ = radical center of the polar-circle-inverses of the power circles

• X₂₃₅ = intersection of tangents to Hatzipolakis-Lozada hyperbola at X₄ and X₁₅₅

• X₂₃₅ = crosspoint, wrt orthic triangle, of X₄ and X₁₅₅

Notes

As a point on the Euler line, X₂₃₅ has Shinagawa coefficients \((F, F - E)\).