X₂₀₄₅

5th EULER-VECTEN-GIBERT POINT

X₂₀₄₅ = inverse-in-orthocentroidal-circle of X₂₀₄₆

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

X₂₀₄₅ (5th 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, 485}

• Related Centers: X₂ (CENTROID), X₃ (CIRCUMCENTER), X₄ (ORTHOCENTER), X₁₇ (1st NAPOLEON POINT), X₄₈₅ (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): \(4.608798366703\)

▸ Linear Combinations1 combination

• X₂₀₄₅ = \(3\) X₂ + \((1 + \sqrt{3})\) X₃

▸ Incident Central Lines9 lines

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

▸ Functional Relations7 relations

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

• X₂₀₄₄ = X₁-of-X₃₈₁PU(5)

• X₂₀₄₄ = {X₂,X₄}-harmonic conjugate of X₂₀₄₃

• X₂₀₄₄ = {X₃,X₄}-harmonic conjugate of X₂₀₄₁

• X₂₀₄₄ = {X₅,X₃₈₁}-harmonic conjugate of X₂₀₄₃

• X₂₀₄₄ = {X₂₀,X₃₇₆}-harmonic conjugate of X₂₀₄₃

• X₂₀₄₄ = {X₃₈₂,X₅₄₉}-harmonic conjugate of X₂₀₄₃

Notes

As a point on the Euler line, X₂₀₄₅ has Shinagawa coefficients \((3 - 3^{1/2},1 - 3^{1/2})\).