X₁₃₂₀

ISOGONAL CONJUGATE OF X₁₃₁₉

X₁₃₂₀ is the isogonal conjugate of X₁₃₁₉.

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

X₁₃₂₀ (ISOGONAL CONJUGATE OF X₁₃₁₉) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {145, 149}

• Related Centers: X₁₄₅ (ANTICOMPLEMENT OF NAGEL POINT), X₁₄₉ (REFLECTION OF X₂₀ IN X₁₀₄)

Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\), where \(f(a,b,c) = (b + c - a)/ (b + c - 2a)\)

Barycentrics: \(af(a,b,c) : bf(b,c,a) : cf(c,a,b)\)

Search Value (6, 9, 13): \(0.778780883938\)

X₁₃₂₀ lies on the Darboux septic.

▸ Incident Central Lines12 lines

{1, 88}, {2, 1000}, {4, 145}, {7, 528}, {8, 11}, {9, 644}, {21, 643}, {80, 519}, {104, 517}, {518, 1156}, {900, 1120}, {1022, 1280}

▸ Functional Relations13 relations

• X₁₃₂₀ = midpoint of X₁₄₅ and X₁₄₉

• X₁₃₂₀ = reflection of X_i in X_j for these (i,j): (8, 11), (100, 1), (1145, 1387)

• X₁₃₂₀ = isogonal conjugate of X₁₃₁₉

• X₁₃₂₀ = anticomplement of X₁₁₄₅

• X₁₃₂₀ = cevapoint of X₁ and X₅₁₇

• X₁₃₂₀ = crosssum of X₉₀₂ and X₁₄₀₄

• X₁₃₂₀ = antigonal conjugate of X₈

• X₁₃₂₀ = symgonal of X₁

• X₁₃₂₀ = trilinear pole of line X₉X₆₅₀

• X₁₃₂₀ = Kirikami concurrent circles image of X₁

• X₁₃₂₀ = polar conjugate of X₃₇₇₉₀

• X₁₃₂₀ = pole wrt polar circle of trilinear polar of X₃₇₇₉₀ (line X₉₀₀X₁₈₄₆)

• X₁₃₂₀ = excentral-to-ABC barycentric image of X₅₅₄₁

Notes

Let DEF be the intouch triangle. Let H_a be the orthocenter of IBC. Let A₁ be the point, other than A, where AI meets the circumcircle. Let \(t_{a}\) be the tangent to the circle (DH_aA₁) at H_a, and define \(t_{b}\) and \(t_{c}\) cyclically. The lines \(t_{a}\), \(t_{b}\) , \(t_{c}\) concur in X₁₃₂₀. (Angel Montesdeoca, September 3, 2020)