X₅₇₆

ORTHOHARMONIC OF X₁₈₇

X₅₇₆ = Brocard-circle-inverse of X₅₇₅

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

X₅₇₆ (ORTHOHARMONIC OF X₁₈₇) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {3, 6}, {23, 184}

• Related Centers: X₃ (CIRCUMCENTER), X₆ (SYMMEDIAN POINT), X₂₃ (FAR-OUT POINT), X₁₈₄ (INVERSE OF X)

Trilinears: \(\sin A + \cot(D/2) \cos A\), where \(\cot(D/2) = - (a^{2} + b^{2} + c^{2})/(12 \operatorname{area})\)

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

Search Value (6, 9, 13): \(-1.901818704500\)

▸ Alternate Coordinates3 forms

• Trilinears: \(3 \sin A - \cos A \cot \omega : :\) (Peter J. C. Moses, 7/20/03)

• Trilinears: \(\cos A - 3 \sin A \tan \omega : :\) (Peter J. C. Moses, 8/22/03)

• Trilinears: \(2 \cos(A + 2\omega ) - \cos(A - 2\omega ) + \cos A : :\)

▸ Linear Combinations1 combination

• X₅₇₆ = X₃ - \(3\) X₆

▸ Incident Central Lines6 lines

{3, 6}, {4, 542}, {5, 524}, {23, 184}, {140, 597}, {262, 385}

▸ Functional Relations11 relations

• X₅₇₆ = reflection of X_i in X_j for these (i,j): (3, 575), (182, 6)

• X₅₇₆ = circumcircle-inverse of X₃₈₂₂₅

• X₅₇₆ = Brocard-circle-inverse of X₅₇₅

• X₅₇₆ = 2nd-Lemoine-circle-nverse of X₁₆₉₁

• X₅₇₆ = isogonal conjugate of X₇₆₀₇

• X₅₇₆ = {X₆₁,X₆₂}-harmonic conjugate of X₃₂

• X₅₇₆ = pole of Lemoine axis wrt circle {X₃₇₁,X₃₇₂,PU(1),PU(39)}

• X₅₇₆ = center of Ehrmann circle

• X₅₇₆ = X₁ of 2nd Ehrmann triangle if ABC is acute

• X₅₇₆ = Cundy-Parry Phi transform of X₁₈₇

• X₅₇₆ = Cundy-Parry Psi transform of X₆₇₁