X₃₅₂₀

EULER LINE INTERCEPT OF LINE X₅₄X_74

X₃₅₂₀ = reflection of X₄ in X₁₅₉₄

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

X₃₅₂₀ (EULER LINE INTERCEPT OF LINE X₅₄X_74) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {4, 1594}, {2, 3} (Euler line)

• Related Centers: X₄ (ORTHOCENTER), X₁₅₉₄ (RIGBY-LALESCU ORTHOPOLE), X₂ (CENTROID), X₃ (CIRCUMCENTER)

Trilinears: \(4 \cos A + \sec A : :\)

Search Value (6, 9, 13): \(13.703857431136\)

▸ Linear Combinations1 combination

• X₃₅₂₀ = \((a^{2} - b^{2} - c^{2})(a^{2} + b^{2} - c^{2})(a^{2} - b^{2} + c^{2})\) X₃ - \(a^{2}b^{2}c^{2}\) X₄

▸ Incident Central Lines13 lines

{2, 3}, {35, 1870}, {39, 112}, {49, 3043}, {54, 74}, {64, 3431}, {93, 930}, {99, 1235}, {184, 3357}, {567, 1986}, {574, 1968}, {578, 1199}, {1970, 3269}

▸ Functional Relations6 relations

• X₃₅₂₀ = reflection of X₄ in X₁₅₉₄

• X₃₅₂₀ = homothetic center of circumorthic triangle and Trinh triangle

• X₃₅₂₀ = X₃₅-of-Trinh-triangle if ABC is acute

• X₃₅₂₀ = Trinh-isogonal conjugate of X₇₆₉₁

• X₃₅₂₀ = exsimilicenter of circumcircle and Trinh circle; the insimilicenter is X₂₀₇₁)

• X₃₅₂₀ = {X₃,X₄}-harmonic conjugate of X₁₈₆

Notes

As a point on the Euler line, X₃₅₂₀ has Shinagawa coefficients \((4F, E - 4F)\).