X₃₆₅₉

FEUERBACH POINT OF EXCENTRAL TRIANGLE

X₃₆₅₉ = X₁₁-of-excentral-triangle

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

X₃₆₅₉ (FEUERBACH POINT OF EXCENTRAL TRIANGLE) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {3, 164}, {55, 258}

• Related Centers: X₃ (CIRCUMCENTER), X₁₆₄ (INCENTER OF EXCENTRAL TRIANGLE), X₅₅ (INSIMILICENTER), X₂₅₈ (CONGRUENT INCIRCLES ISOSCELIZER POINT)

Trilinears: \(f(A,B,C) : f(B,C,A) : f(C,A,B)\), where \(f(A,B,C) = (\cos(A/2))/(\sin(B/2) - \sin(C/2))\)

Barycentrics: \(a^{2}/(\csc(B/2) - \csc(C/2)) : :\)

Search Value (6, 9, 13): \(1.526642190071\)

X₃₆₅₉ lies on the circumcircle.

▸ Incident Central Lines4 lines

{3, 164}, {55, 258}, {106, 1130}, {759, 1128}

▸ Functional Relations6 relations

• X₃₆₅₉ = X₁₁-of-excentral-triangle

• X₃₆₅₉ = X₁₁₉-of-hexyl-triangle

• X₃₆₅₉ = center of hyperbola {{X₁,X₁₆₄,X₁₆₆,X₁₆₇,X₁₆₈,excenters}}

• X₃₆₅₉ = X₁₀₀ of 1st circumperp triangle

• X₃₆₅₉ = Λ(X₆₇₂₈, X₁₀₄₉₂)

• X₃₆₅₉ = isogonal conjugate of infinite point of antiorthic axis of intouch triangle (or excentral triangle)

Notes

Let I be the incenter and I_A the A-excenter of the triangle ABC. Let L_A be the line joining the circumcenter of triangle BCI and the incenter of triangle BCI_A, and define L_B and L_C cyclically. The lines L_A, L_B, L_C concur in X₃₆₅₉. (Seiichi Kirikami, April 12, 2010)

Let E_a be the ellipse with B and C as foci and passing through X₁, and define E_b and E_c cyclically. Let L_a be the line tangent to E_a at X₁, and define L_b and L_c cyclically. Let A' = L_a∩BC, B' = L_b∩CA, C' = L_c∩AB. Then A', B', C' are collinear, and the line A'B'C' meets the line at infinity at the isogonal conjugate of X₃₆₅₉. (Randy Hutson, April 9, 2016)

Let I_AI_BI_C be the excentral triangle. The lines X₁X₃ of triangles BCI_A, CAI_B, ABI_C concur in X₃₆₅₉. (Randy Hutson, June 7, 2019)