X₂₀₅₂

ISOGONAL CONJUGATE OF X₅₇₇

X₂₀₅₂ is the isogonal conjugate of X₅₇₇.

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

X₂₀₅₂ (ISOGONAL CONJUGATE OF X₅₇₇) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {4, 51}, {24, 96}

• Related Centers: X₅₇₇ (HARMONIC OF X₂₁₆), X₄ (ORTHOCENTER), X₅₁ (CENTROID OF ORTHIC TRIANGLE), X₂₄ (PERSPECTOR OF ABC AND ORTHIC-OF-ORTHIC TRIANGLE), X₉₆ (ISOGONAL CONJUGATE OF X₅₂)

Trilinears: \(\csc A \sec^{2} A : :\)

Barycentrics: \(\sec^{2} A : :\)

Search Value (6, 9, 13): \(1.329869583541\)

▸ Alternate Coordinates3 forms

• Trilinears: \(\sec A \csc 2A : :\)

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

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

▸ Incident Central Lines22 lines

{2, 216}, {4, 51}, {6, 275}, {10, 158}, {13, 473}, {14, 472}, {17, 470}, {18, 471}, {24, 96}, {25, 98}, {63, 1947}, {76, 297}, {83, 458}, {92, 226}, {184, 436}, {262, 427}, {331, 1446}, {394, 801}, {401, 1968}, {485, 1585}, {486, 1586}, {648, 1993}

▸ Functional Relations10 relations

• X₂₀₅₂ = isogonal conjugate of X₅₇₇

• X₂₀₅₂ = isotomic conjugate of X₃₉₄

• X₂₀₅₂ = cevapoint of X_i and X_j for these (i,j): (4, 393), (6, 24), (324, 467), (1585, 1586)

• X₂₀₅₂ = X_i-cross conjugate of X_j for these (i,j): (4, 264), (6, 847), (53, 4), (393, 1093), (686, 1304)

• X₂₀₅₂ = trilinear pole of line X₄₀₃X₅₂₃ (polar of X₃ wrt polar circle)

• X₂₀₅₂ = pole wrt polar circle of trilinear polar of X₃ (line X₅₂₀X₆₄₇)

• X₂₀₅₂ = polar conjugate of X₃

• X₂₀₅₂ = X₄₃-of-orthic-triangle if ABC is acute

• X₂₀₅₂ = complement of isotomic conjugate of X₃₄₂₈₇

• X₂₀₅₂ = barycentric product X₁₀₇*X₈₅₀

Notes

Let H_aH_bH_c be the orthic triangle of a triangle ABC, and let

• O(A) = circle with center A and radius AH_a, and define O(B) and O(C) cyclically

• p(A) = polar of X₄ wrt O(A), and define p(B) and p(C) cyclically

• A″ = p(B)∩p(C), and define B″ and C″ cyclically.

Then A″B″C″ is homothetic to ABC, and the center of homothety is X₂₀₅₂. (Angel Montesdeoca, September 30, 2016)