X₃₀₉₀

INTERSECTION OF LINES X₂X_3 AND X₁₁X_1058

X₃₀₉₀ = nine-point-circle-inverse of radical trace of Grebe circle and de Longchamps circle

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

X₃₀₉₀ (INTERSECTION OF LINES X₂X_3 AND X₁₁X_1058) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 3} (Euler line), {69, 576}

• Related Centers: X₂ (CENTROID), X₃ (CIRCUMCENTER), X₄ (ORTHOCENTER), X₆₉ (SYMMEDIAN POINT OF THE ANTICOMPLEMENTARY TRIANGLE), X₅₇₆ (ORTHOHARMONIC OF X₁₈₇)

Trilinears: \(\sin B \sin C + \cos(B - C) : :\)

Barycentrics: \(a^{4} - 4a^{2}(b^{2} + c^{2}) + 3(b^{2} - c^{2})^{2} : :\)

Search Value (6, 9, 13): \(1.442799049144\)

▸ Alternate Coordinates4 forms

• Trilinears: \(2 \cos A + 3 \cos B \cos C : :\)

• Trilinears: \(3 \sec A + 2 \sec B \sec C : :\)

• Barycentrics: \(2 S^{2} + S_{B} S_{C} : :\)

• Barycentrics: \(\cot B \cot C + 2 : :\)

▸ Incident Central Lines22 lines

{2, 3}, {11, 1058}, {12, 1056}, {69, 576}, {76, 1007}, {110, 569}, {155, 1199}, {182, 1614}, {233, 1249}, {373, 389}, {388, 499}, {485, 3069}, {486, 3068}, {497, 498}, {575, 1352}, {590, 1588}, {615, 1587}, {748, 3072}, {750, 3073}, {944, 1125}, {946, 1698}, {1493, 2888}

▸ Functional Relations17 relations

• X₃₀₉₀ = reflection of X_i in X_j for these (i,j): (3523, 3526), (3528, 3523)

• X₃₀₉₀ = isotomic conjugate of X₃₆₉₄₈

• X₃₀₉₀ = complement of X₃₅₂₃

• X₃₀₉₀ = anticomplement of X₃₅₂₆

• X₃₀₉₀ = circumcircle-inverse of X₃₇₉₃₉

• X₃₀₉₀ = nine-point-circle-inverse of anticomplement of X₃₇₉₇₁

• X₃₀₉₀ = nine-point-circle-inverse of complement of X₃₇₉₄₅

• X₃₀₉₀ = nine-point-circle-inverse of radical trace of Grebe circle and de Longchamps circle

• X₃₀₉₀ = polar conjugate of isogonal conjugate of X₃₆₇₅₁

• X₃₀₉₀ = inverse-in-orthocentroidal-circle of X₆₃₁

• X₃₀₉₀ = homothetic center of orthocentroidal triangle and X₅-Brocard triangle

• X₃₀₉₀ = homothetic center of anti-Euler triangle and cross-triangle of ABC and Euler triangle

• X₃₀₉₀ = homothetic center of Euler triangle and cross-triangle of ABC and anti-Euler triangle

• X₃₀₉₀ = homothetic center of ABC and cross-triangle of Euler and anti-Euler triangles

• X₃₀₉₀ = homothetic center of X₂-altimedial and X₃-anti-altimedial triangles

• X₃₀₉₀ = homothetic center of X₄-altimedial and X₂₀-anti-altimedial triangles

• X₃₀₉₀ = {X_i,X_j}-harmonic conjugate of X_k for these (i,j,k): (2, 3, 3525), (2, 4, 631), (2, 5, 4), (2, 20, 140), (2, 384, 32970), (3, 4, 3529), (3, 5, 3091), (4, 5, 3545), (4, 20, 15682), (4, 3544, 3091), (76, 1007, 32818), (373, 389, 15024)

Notes

As a point on the Euler line, X₃₀₉₀ has Shinagawa coefficients \((2,1)\).