X₅₇₇

HARMONIC OF X₂₁₆

X₅₇₇ = complement of X₃₁₇

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

X₅₇₇ (HARMONIC OF X₂₁₆) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {3, 6}, {30, 53}

• Related Centers: X₂₀₅₂ (ISOGONAL CONJUGATE OF X₅₇₇), X₃ (CIRCUMCENTER), X₆ (SYMMEDIAN POINT), X₃₀ (EULER INFINITY POINT), X₅₃ (SYMMEDIAN POINT OF ORTHIC TRIANGLE)

Trilinears: \(\sin A \cos^{2} A : :\)

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

Search Value (6, 9, 13): \(1.971561272069\)

▸ Alternate Coordinates6 forms

• Trilinears: \(\cos A (\tan A + \tan B + \tan C) - \sin A : :\)

• Trilinears: \(\cos A - \sin A (\cot A \cot B \cot C) : :\)

• Trilinears: \(\cos A \sin 2A : :\)

• Trilinears: \(\cos(A - T) : :\), T as at X₃₈₉

• Trilinears: \(\sec B \sec C - \csc B \csc C : :\)

• Barycentrics: \(\sin^{2} 2A : :\)

▸ Incident Central Lines18 lines

{2, 95}, {3, 6}, {20, 393}, {22, 232}, {30, 53}, {48, 603}, {69, 248}, {112, 376}, {141, 441}, {160, 206}, {172, 1038}, {184, 418}, {198, 478}, {219, 906}, {220, 268}, {264, 401}, {395, 466}, {396, 465}

▸ Functional Relations22 relations

• X₅₇₇ = reflection of X₃₆₄₂₆ in X₂₃₅₈₃

• X₅₇₇ = isogonal conjugate of X₂₀₅₂

• X₅₇₇ = isotomic conjugate of X₁₈₀₂₇

• X₅₇₇ = complement of X₃₁₇

• X₅₇₇ = X_i-Ceva conjugate of X_j for these (i,j): (3, 184), (97, 3)

• X₅₇₇ = X₄₁₈-cross conjugate of X₃

• X₅₇₇ = crosspoint of X_i and X_j for these (i,j): (2, 68), (3, 394)

• X₅₇₇ = crosssum of X_i and X_j for these (i,j): (4, 393), (6, 24), (324, 467)

• X₅₇₇ = crossdifference of every pair of points on line X₄₀₃X₅₂₃

• X₅₇₇ = inverse-in-Brocard-circle of X₂₁₆

• X₅₇₇ = perspector of circumconic centered at X₁₁₄₇

• X₅₇₇ = center of circumconic that is locus of trilinear poles of lines passing through X₁₁₄₇

• X₅₇₇ = X₂-Ceva conjugate of X₁₁₄₇

• X₅₇₇ = X₉₂-isoconjugate of X₄

• X₅₇₇ = X₁₅₇₇-isoconjugate of X₁₀₇

• X₅₇₇ = {X₆,X₅₀}-harmonic conjugate of X₅₇₁

• X₅₇₇ = barycentric square of X₃

• X₅₇₇ = orthic-to-ABC barycentric image of X₅₃

• X₅₇₇ = antipode of X₃₆₄₂₆ in barycentric Euler inellipse

• X₅₇₇ = circle-{X₃₇₁,X₃₇₂,PU(1),PU(39)}}-inverse of X₃₈₉

• X₅₇₇ = {X₃₇₁,X₃₇₂}-harmonic conjugate of X₃₈₉

• X₅₇₇ = homothetic center of [mid-triangle of 1st & 2nd Kenmotu diagonals triangles] and [mid-triangle of orthic and dual of orthic triangles]