X₁₅₇₇

ISOGONAL CONJUGATE OF X₁₆₃

X₁₅₇₇ is the isogonal conjugate of X₁₆₃.

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

X₁₅₇₇ (ISOGONAL CONJUGATE OF X₁₆₃) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {240, 522}, {1, 810}

• Related Centers: X₁₆₃ (TRILINEAR PRODUCT X₆*X₁₁₀), X₂₄₀ (-LINE CONJUGATE OF X₄₈), X₅₂₂ (ISOGONAL CONJUGATE OF X₁₀₉), X₁ (INCENTER), X₈₁₀ (CROSSPOINT OF X₁ AND X₁₆₃)

Trilinears: \(b^{2}c^{2}(b^{2} - c^{2}) : c^{2}a^{2}(c^{2} - a^{2}) : a^{2}b^{2}(a^{2} - b^{2})\)

Barycentrics: \(bc(b^{2} - c^{2}) : ca(c^{2} - a^{2}) : ab(a^{2} - b^{2})\)

Search Value (6, 9, 13): \(34.530588121765\)

▸ Alternate Coordinates4 forms

• Trilinears: \((\csc 2A)(\sin 2B - \sin 2C) : :\)

• Trilinears: \(d(a,b,c) : :\), where d(a,b,c) = directed distance from A to Brocard axis

• Trilinears: \(|AP(1)|^{2} - |AU(1)|^{2} : :\)

• Barycentrics: \(\sin(B - C) : \sin(C - A) : \sin(A - B)\)

▸ Incident Central Lines9 lines

{1, 810}, {115, 1111}, {163, 811}, {240, 522}, {514, 661}, {667, 814}, {784, 149}, {798, 812}, {826, 1089}

▸ Functional Relations21 relations

• X₁₅₇₇ = isogonal conjugate of X₁₆₃

• X₁₅₇₇ = isotomic conjugate of X₆₆₂

• X₁₅₇₇ = X_i-Ceva conjugate of X_j for these (i,j): (75, 1109), (76, 1111), (693, 523), (799, 75), (811, 1), (823, 92)

• X₁₅₇₇ = cevapoint of X₆₅₆ and X₆₆₁

• X₁₅₇₇ = X_i-cross conjugate of X_j for these (i,j): (115, 1089), (1109, 75)

• X₁₅₇₇ = crosspoint of X_i and X_j for these (i,j): (75, 799), (82, 162), (92, 823), (662, 2167), (811, 1969), (1240, 1978)

• X₁₅₇₇ = crosssum of X_i and X_j for these (i,j): (31, 798), (38, 656), (48, 822), (649, 2260), (652, 2269), (661, 1953), (1923, 1924)

• X₁₅₇₇ = crossdifference of every pair of points on line X₃₁X₄₈

• X₁₅₇₇ = X_i-aleph conjugate of X_j for these (i,j): (648, 656), (811, 1969)

• X₁₅₇₇ = complement of X₄₅₆₀

• X₁₅₇₇ = bicentric difference of PU(14)

• X₁₅₇₇ = PU(14)-harmonic conjugate of X₁₉₃₀

• X₁₅₇₇ = trilinear product of PU(40)

• X₁₅₇₇ = perspector of hyperbola {{A, B, C,X₇₅,X₉₂}} (centered at X₄₈₅₈)

• X₁₅₇₇ = center of circumconic that is locus of trilinear poles of lines passing through X₄₈₅₈

• X₁₅₇₇ = X₂-Ceva conjugate of X₄₈₅₈

• X₁₅₇₇ = X₆-isoconjugate of X₁₁₀

• X₁₅₇₇ = pole wrt polar circle of trilinear polar of X₁₆₂ (line X₁X₁₉)

• X₁₅₇₇ = X₄₈-isoconjugate (polar conjugate) of X₁₆₂

• X₁₅₇₇ = X₅₂-isoconjugate of X₃₂₆₉₂

• X₁₅₇₇ = trilinear pole of line X₁₁₀₉X₂₆₃₂

Notes

Let J_aJ_bJ_c be the excentral triangle, and let U be the bianticevian conic of X₁ and X₄. Let T be the tangential triangle, wrt the anticevian triangle of X₁₉, of U. Then J_aJ_bJ_c and T are perspective, and their perspector is X₁₅₇₇. (Randy Hutson, December 26, 2015)

X₁₅₇₇ is the trilinear multiplier for the Kiepert hyperbola. (The trilinear product of X₁₅₇₇ and the circumcircle is the Kiepert hyperbola.) (Randy Hutson, August 19, 2019)