X₃₈₃

EULER LINE INTERCEPT OF LINE X₁₄X_98

X₃₈₃ = reflection of X₁₀₈₀ in X₁₅₁₃

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

X₃₈₃ (EULER LINE INTERCEPT OF LINE X₁₄X_98) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {1080, 1513}, {2, 3} (Euler line)

• Related Centers: X₁₀₈₀ (INTERCEPT OF EULER LINE AND LINE), X₁₅₁₃ (ORTHOJOIN OF X₂), X₂ (CENTROID), X₃ (CIRCUMCENTER)

Trilinears: \(f(A,B,C) : f(B,C,A) : f(C,A,B)\), where \(f(A,B,C) = \csc(B - C) [\sin 2B \cos(C - \omega ) \sin(C + \pi /3) - \sin 2C \cos(B - \omega ) \sin(B + \pi /3)]\)

Barycentrics: \((\sin A)f(A,B,C) : (\sin B)f(B,C,A) : (\sin C)f(C,A,B)\)

Search Value (6, 9, 13): \(-16.834262678779\)

▸ Incident Central Lines6 lines

{2, 3}, {13, 262}, {14, 98}, {183, 621}, {299, 511}, {325, 622}

▸ Functional Relations3 relations

• X₃₈₃ = reflection of X₁₀₈₀ in X₁₅₁₃

• X₃₈₃ = inverse-in-orthocentroidal-circle of X₁₀₈₀

• X₃₈₃ = orthoptic-circle-of-Steiner-inellipse-inverse of X₃₂₄₆₀

Notes

As a point on the Euler line, X₃₈₃ has Shinagawa coefficients \((S, -3^{1/2}(E + F))\).