X₃₉₄

X₆₉-CEVA CONJUGATE OF X₃

X₃₉₄ = isogonal conjugate of X₃₉₃

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

X₃₉₄ (X₆₉-CEVA CONJUGATE OF X₃) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 6}, {3, 49}

• Related Centers: X₆₉ (SYMMEDIAN POINT OF THE ANTICOMPLEMENTARY TRIANGLE), X₃ (CIRCUMCENTER), X₂ (CENTROID), X₆ (SYMMEDIAN POINT), X₄₉ (CENTER OF SINE-TRIPLE-ANGLE CIRCLE)

Trilinears: \(\cos A \cot A : \cos B \cot B : \cos C \cot C\)

Barycentrics: \(\cos^{2} A : \cos^{2} B : \cos^{2} C\)

Search Value (6, 9, 13): \(3.880761683324\)

▸ Alternate Coordinates4 forms

• Trilinears: \(\sin A - \csc A : :\)

• Trilinears: \(a \cot^{2} A : :\)

• Barycentrics: \(1 - \tan B \tan C : :\)

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

▸ Incident Central Lines23 lines

{2, 6}, {3, 49}, {20, 1032}, {22, 110}, {25, 511}, {63, 77}, {72, 1060}, {76, 275}, {78, 271}, {287, 305}, {297, 315}, {329, 651}, {393, 837}, {399, 541}, {470, 633}, {471, 634}, {472, 622}, {473, 621}, {493, 1504}, {494, 1505}, {611, 612}, {613, 614}, {1062, 1069}

▸ Functional Relations17 relations

• X₃₉₄ = isogonal conjugate of X₃₉₃

• X₃₉₄ = isotomic conjugate of X₂₀₅₂

• X₃₉₄ = complement of X₆₅₁₅

• X₃₉₄ = crosssum of X₅₄₁₂ and X₅₄₁₃

• X₃₉₄ = crosspoint of X₆ and X₁₄₉₈ wrt both the excentral and tangential triangles

• X₃₉₄ = X₆₉-Ceva conjugate of X₃

• X₃₉₄ = crosspoint of X₄₉₃ and X₄₉₄

• X₃₉₄ = crosspoint of X₁₁₀₉₀ and X₁₁₀₉₁

• X₃₉₄ = crosssum of X₄ and X₄₅₉

• X₃₉₄ = crossdifference of every pair of points on line X₄₆₀X₅₁₂

• X₃₉₄ = intersection of tangents to hyperbola {A, B, C,X₂,X₆}} at X₄₉₃ and X₄₉₄

• X₃₉₄ = Danneels point of X₆₉

• X₃₉₄ = pole wrt polar circle of trilinear polar of X₁₀₉₃

• X₃₉₄ = X₄₈-isoconjugate (polar conjugate) of X₁₀₉₃

• X₃₉₄ = X₉₂-isoconjugate of X₂₅

• X₃₉₄ = intersection of tangents at X₂ and X₂₀ to Lucas cubic K007

• X₃₉₄ = barycentric square of X₆₃

Notes

Let A' be the trilinear pole of the tangent to the circumcircle at the antipode of A, and define B' and C' cyclically. The lines AA', BB', CC' concur in X₃₉₄. (Randy Hutson, November 18, 2015)

Let H_aH_bH_c be the orthic triangle. Let A″ be the crosssum of the (real or imaginary) circumcircle intercepts of line H_bH_c. Define B″ andC″ cyclically. The lines AA″, BB″, CC″ concur in X₃₉₃. (Randy Hutson, July 31 2018)