X₄₇₅

EULER LINE INTERCEPT OF LINE X₁₀X_34

X₄₇₅ = inverse-in-orthocentroidal-circle of X₄₀₆

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

X₄₇₅ (EULER LINE INTERCEPT OF LINE X₁₀X_34) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 3} (Euler line), {10, 34}

• Related Centers: X₂ (CENTROID), X₃ (CIRCUMCENTER), X₄ (ORTHOCENTER), X₁₀ (SPIEKER CENTER), X₃₄ (-BETH CONJUGATE OF X₄)

Trilinears: \(\sec A - (a + b + c)/a : \sec B - (a + b + c)/b : \sec C - (a + b + c)/c\)

Barycentrics: \(a \sec A - (a + b + c) : b \sec B - (a + b + c) : c \sec C - (a + b + c)\)

Search Value (6, 9, 13): \(1.816163529168\)

▸ Incident Central Lines5 lines

{2, 3}, {8, 1063}, {10, 34}, {264, 274}, {318, 1068}

▸ Functional Relations2 relations

• X₄₇₅ = inverse-in-orthocentroidal-circle of X₄₀₆

• X₄₇₅ = polar conjugate of isogonal conjugate of X₃₆₇₄₃

Notes

As a point on the Euler line, X₄₇₅ has Shinagawa coefficients \((\$a\$F,-abc)\).