X₂₀₄₁

1st EULER-VECTEN-GIBERT POINT

X₂₀₄₁ = reflection of X₂₀₄₂ in X₃₈₂

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

X₂₀₄₁ (1st EULER-VECTEN-GIBERT POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2042, 382}, {2, 3} (Euler line)

• Related Centers: X₂₀₄₂ (2nd EULER-VECTEN-GIBERT POINT), X₃₈₂ (REFLECTION OF CIRCUMCENTER IN ORTHOCENTER), X₂ (CENTROID), X₃ (CIRCUMCENTER)

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

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

Search Value (6, 9, 13): \(-39.715192079124\)

▸ Alternate Coordinates2 forms

• Trilinears: \(g(A,B,C) : g(B,C,A) : g(C,A,B)\), where \(g(A,B,C) = \sin(B + \pi /4)\sin(C + \pi /4) - \cos(A - \pi /6)\) (M. Iliev, 5/13/07)

• Trilinears: \(h(A,B,C) : h(B,C,A) : h(C,A,B)\), where \(h(A,B,C) = \sin A + \cos(B - C) - 2 \cos(A - \pi /6)\) (M. Iliev, 5/13/07)

▸ Linear Combinations1 combination

• X₂₀₄₁ = \(3\) X₂ - \((1 + \sqrt{3})\) X₃

▸ Incident Central Lines7 lines

{2, 3}, {15, 485}, {16, 486}, {489, 622}, {490, 621}, {491, 628}, {492, 627}

▸ Functional Relations2 relations

• X₂₀₄₁ = reflection of X₂₀₄₂ in X₃₈₂

• X₂₀₄₁ = {X_i,X_j}-harmonic conjugate of X_k for these (i,j,k): (2, 641, 2042), (3, 4, 2044), (3, 5, 2042), (4, 20, 2042), (376, 3832, 2042), (381, 548, 2042)

Notes

As a point on the Euler line, X₂₀₄₁ has Shinagawa coefficients \((3^{1/2} - 1, -3^{1/2} - 1)\).