X₁₉₈₉

ISOGONAL CONJUGATE OF X₃₂₃

X₁₉₈₉ plays a major role in the theory of special isocubics, as presented in Chapter 6 of Jean-Pierre Ehrmann and Bernard Gibert,, “Special Isocubics in the Triangle Plane,” downloadable from Bernard Gibert, Cubics in the Triangle Plane.

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

X₁₉₈₉ (ISOGONAL CONJUGATE OF X₃₂₃) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Derived Triangle: Medial Triangle △M_aM_bM_c — vertices M_a, M_b, M_c (Formed by the midpoints of sides BC, CA, AB)

• Central Lines: {2, 1272}

• Related Centers: X₂ (CENTROID), X₁₂₇₂ (ISOTOMIC CONJUGATE OF X₁₁₃₈)

Trilinears: \(\sin^{2} A \csc 3A : :\)

Barycentrics: \(\sin^{3} A \csc 3A : \sin^{3} B \csc 3B : \sin^{3} C \csc 3C\)

Search Value (6, 9, 13): \(-0.053663225028\)

▸ Alternate Coordinates3 forms

• Trilinears: \(a/(1 - 4 \cos^{2} A) : :\)

• Trilinears: \(\csc(A + \pi /3) + \csc(A - \pi /3) : :\)

• Barycentrics: \(1/((a^{2} - b^{2} - c^{2})^{2} - b^{2} c^{2}) : :\)

▸ Incident Central Lines8 lines

{2, 94}, {6, 13}, {30, 50}, {53, 112}, {67, 868}, {111, 230}, {403, 1990}, {1427, 2006}

▸ Functional Relations20 relations

• X₁₉₈₉ = isogonal conjugate of X₃₂₃

• X₁₉₈₉ = complement of X₁₂₇₂

• X₁₉₈₉ = X₉₄-Ceva conjugate of X₂₆₅

• X₁₉₈₉ = cevapoint of X_i and X_j for these (i,j): (53, 1990), (115, 1637), (395, 396)

• X₁₉₈₉ = crosspoint of X₂ and X₁₁₃₈

• X₁₉₈₉ = crosssum of X₆ and X₃₉₉

• X₁₉₈₉ = barycentric product of X₁₃ and X₁₄

• X₁₉₈₉ = isotomic conjugate of X₇₇₉₉

• X₁₉₈₉ = inverse-in-Kiepert-hyperbola of X₂₆₅

• X₁₉₈₉ = {X₁₃,X₁₄}-harmonic conjugate of X₂₆₅

• X₁₉₈₉ = trilinear pole of line X₅₁X₅₁₂

• X₁₉₈₉ = pole wrt polar circle of trilinear polar of X₃₄₀

• X₁₉₈₉ = X₄₈-isoconjugate (polar conjugate) of X₃₄₀

• X₁₉₈₉ = X₅₀-of-orthocentroidal-triangle

• X₁₉₈₉ = perspector of ABC and unary cofactor triangle of Trinh triangle

• X₁₉₈₉ = circumcircle-inverse of X₁₅₅₅₀

• X₁₉₈₉ = cevapoint of X₃₆₂₉₈ and X₃₆₂₉₉

• X₁₉₈₉ = barycentric product X₇₉*X₈₀

• X₁₉₈₉ = barycentric product of circumcircle intercepts of Johnson circle (or line PU(5), X₅X₅₂₃)

• X₁₉₈₉ = Dao-Moses-Telv-circle-inverse of X₃₄₃₁₀

Notes

X₁₉₈₉ is the barycentric product X₁₃*X₁₄ of the Fermat points. The line through X₅₀ parallel to the Eular line X₂X₃ passes through X₁₉₈₉.

Let A₁B₁C₁ and A₂B₂C₂ be the 1st and 2nd Ehrmann circumscribing triangles. X₁₉₈₉ is the barycentric product A₁*A₂ = B₁*B₂ = C₁*C₂. (Randy Hutson, June 27, 2018)

Let V_aV_bV_c be the Ehrmann vertex-triangle. Let A' be the barycentric product V_b*V_c, and define B', C' cyclically. The lines AA', BB', CC' concur in X₁₉₈₉. (Randy Hutson, June 27, 2018)

Let V_aV_bV_c and SaSbSc be the Ehrmann vertex-triangle and Ehrmann side-triangle, resp. Let A' be the barycentric product V_a*S_a, and define B', C' cyclically. The lines AA', BB', CC' concur in X₁₉₈₉. (Randy Hutson, June 27, 2018)