X₃₀₅₉

INTERSECTION OF LINES X₇X_8 AND X₉X_55

X₃₀₅₉ = cevapoint of X₁ and X₂₉₄₂

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

X₃₀₅₉ (INTERSECTION OF LINES X₇X_8 AND X₉X_55) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {7, 8}, {9, 55}

• Related Centers: X₁ (INCENTER), X₂₉₄₂ (EXCENTRAL-ISOGONAL CONJUGATE OF X₇), X₇ (GERGONNE POINT), X₈ (NAGEL POINT), X₉ (MITTENPUNKT), X₅₅ (INSIMILICENTER)

Trilinears: \((b + c - a)^{2}(b^{2} + c^{2} - 2bc -ab - ac) : :\)

Search Value (6, 9, 13): \(4.721924024053\)

X₃₀₅₉ lies on the Mandart hyperbola.

▸ Incident Central Lines7 lines

{7, 8}, {9, 55}, {37, 2340}, {72, 516}, {78, 1001}, {142, 354}, {971, 2951}

▸ Functional Relations8 relations

• X₃₀₅₉ = reflection of X_i in X_j for these (i,j): (65, 2550), (390, 960), (6068, 14740)

• X₃₀₅₉ = cevapoint of X₁ and X₂₉₄₂

• X₃₀₅₉ = crosspoint of X₈ and X₂₀₀

• X₃₀₅₉ = crosssum of X_i and X_j for these (i,j): (56, 269), (57, 1617)

• X₃₀₅₉ = complement of X₃₀₆₂₈

• X₃₀₅₉ = excentral-to-ABC barycentric image of X₇

• X₃₀₅₉ = antipode of X₆₀₆₈ in the Mandart hyperbola

• X₃₀₅₉ = extouch-isogonal conjugate of X₂₀₅₈₈

Notes

Let A'B'C' be the intouch triangle of the extangents triangle, if ABC is acute. Then A'B'C' is perspective to the extouch triangle at X₃₀₅₉. (Randy Hutson, December 2, 2017)