X₄₀₇

CROSSPOINT OF X₄ AND X₂₂₅

X₄₀₇ is the crosspoint of X₄ and X₂₂₅.

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

X₄₀₇ (CROSSPOINT OF X₄ AND X₂₂₅) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 3} (Euler line), {12, 228}

• Related Centers: X₄ (ORTHOCENTER), X₂₂₅ (-CEVA CONJUGATE OF X₆₅), X₂ (CENTROID), X₃ (CIRCUMCENTER), X₁₂ ({X), X₂₂₈ (-CEVA CONJUGATE OF X₇₁)

Trilinears: \((v + w)\sec A : (w + u)\sec B : (u + v)\sec C\), where \(u : v : w\) = X₂₁; e.g., \(u = u(A,B,C) = 1/(\cos B + \cos C)\)

Barycentrics: \((v + w) \tan A : (w + u) \tan B : (u + v) \tan C\)

Search Value (6, 9, 13): \(-1.726341097973\)

▸ Incident Central Lines4 lines

{2, 3}, {12, 228}, {65, 225}, {117, 136}

▸ Functional Relations3 relations

• X₄₀₇ = crosspoint of X₄ and X₂₂₅

• X₄₀₇ = crosssum of X_i and X_j for these (i,j): (3, 283), (21, 411)

• X₄₀₇ = tangential-to-orthic similarity image of X₃₁₄₅

Notes

As a point on the Euler line, X₄₀₇ has Shinagawa coefficients \((FS^{2}, -(\$aS_{A}\$)^{2} - FS^{2})\).