X₂₀₄₆

6th EULER-VECTEN-GIBERT POINT

X₂₀₄₆ = inverse-in-orthocentroidal-circle of X₂₀₄₅

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

X₂₀₄₆ (6th EULER-VECTEN-GIBERT POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 3} (Euler line), {17, 486}

• Related Centers: X₂ (CENTROID), X₃ (CIRCUMCENTER), X₄ (ORTHOCENTER), X₁₇ (1st NAPOLEON POINT), X₄₈₆ (INNER VECTEN POINT)

Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\), where \(f(a,b,c) = bc[- 2(b^{2} - c^{2})^{2} + a^{2}(3b^{2} + 3c^{2} - a^{2}) - 3^{1/2}[(b^{2} - c^{2})^{2} - a^{2}(b^{2} + c^{2})]]\)

Barycentrics: \(af(a,b,c) : bf(b,c,a) : cf(c,a,b)\)

Search Value (6, 9, 13): \(1.288861387768\)

▸ Linear Combinations1 combination

• X₂₀₄₆ = \(3\) X₂ + \((1 - \sqrt{3})\) X₃

▸ Incident Central Lines9 lines

{2, 3}, {17, 486}, {18, 485}, {302, 638}, {303, 637}, {397, 615}, {398, 590}, {491, 633}, {492, 634}

▸ Functional Relations1 relation

• X₂₀₄₆ = inverse-in-orthocentroidal-circle of X₂₀₄₅

Notes

As a point on the Euler line, X₂₀₄₆ has Shinagawa coefficients \((3 + 3^{1/2},1 + 3^{1/2})\).