X₉₂₇

ISOGONAL CONJUGATE OF X₉₂₆

X₉₂₇ is the isogonal conjugate of X₉₂₆.

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

X₉₂₇ (ISOGONAL CONJUGATE OF X₉₂₆) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {101, 514}, {103, 516}

• Related Centers: X₉₂₆ (CROSSDIFFERENCE OF X₆ AND X₇), X₁₀₁ (Ψ(INCENTER, SYMMEDIAN POINT)), X₅₁₄ (ISOGONAL CONJUGATE OF X₁₀₁), X₁₀₃ (ANTIPODE OF X₁₀₁), X₅₁₆ (ISOGONAL CONJUGATE OF X₁₀₃)

Trilinears: \(1/(bw - cv) : :\), where \(u : v : w\) = X₇

Search Value (6, 9, 13): \(-0.430811911328\)

X₉₂₇ lies on the circumcircle.

▸ Incident Central Lines6 lines

{7, 840}, {100, 693}, {101, 514}, {103, 516}, {109, 658}, {813, 1025}

▸ Functional Relations9 relations

• X₉₂₇ = isogonal conjugate of X₉₂₆

• X₉₂₇ = trilinear pole of line X₆X₇

• X₉₂₇ = Ψ(X₁, X₈₅)

• X₉₂₇ = Ψ(X₆, X₇)

• X₉₂₇ = Ψ(X₄₁, X₁)

• X₉₂₇ = Ψ(X₆₅₀, X₂)

• X₉₂₇ = circumcircle-intercept, other than A, B, C, of conic {A, B, C,PU(57)}

• X₉₂₇ = circumcircle-intercept, other than A, B, C, of conic {A, B, C,PU(96)}}

• X₉₂₇ = intersection of antipedal lines of X₁₀₁ and X₁₀₃

Notes

Let L_a, L_b, L_c be the lines through A, B, C, resp. parallel to the Gergonne line. Let M_a, M_b, M_c be the reflections of BC, CA, AB in L_a, L_b, L_c, resp. Let A' = M_b∩M_c, and define B' and C' cyclically. Triangle A'B'C' is inversely similar to, and 3 times the size of, ABC. The centroid of A'B'C' is X₉₂₇; see Hyacinthos #16741/16782, Sep 2008. (Randy Hutson, March 25, 2016)

Let A', B', C' be the intersections of line X₇X₈ and lines BC, CA, AB, resp. The circumcircles of AB'C', BC'A', CA'B' concur in X₉₂₇. (Randy Hutson, March 25, 2016)