X₁₀₈₀

INTERCEPT OF EULER LINE AND LINE (13)X₉₈

Coordinates for X₁₀₈₀ are obtained from those of X₃₈₃ by changing π/3 to - π/3; contributed by Edward Brisse.

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

X₁₀₈₀ (INTERCEPT OF EULER LINE AND LINE (13)X₉₈) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 3} (Euler line), {13, 98}

• Related Centers: X₂ (CENTROID), X₃ (CIRCUMCENTER), X₄ (ORTHOCENTER), X₁₃ (1st ISOGONIC CENTER), X₉₈ (TARRY POINT)

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): \(-2.650120542076\)

▸ Alternate Coordinates1 form

• Trilinears: \(g(a,b,c) : g(b,c,a) : g(c,a,b)\), where \(g(a,b,c) = bc[3(a^{2} + b^{2} + c^{2})(a^{2} - b^{2} + c^{2})(a^{2} + b^{2} - c^{2}) + 64(3^{1/2})\sigma ^{3}]\) (M. Iliev, 4/12/07)

▸ Incident Central Lines6 lines

{2, 3}, {13, 98}, {14, 262}, {183, 622}, {298, 511}, {325, 621}

▸ Functional Relations1 relation

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

Notes

As a point on the Euler line, X₁₀₈₀ has Shinagawa coefficients \((3^{1/2}S, 3E + 3F)\).