X₁₆₄

INCENTER OF EXCENTRAL TRIANGLE

X₁₆₄ is the incenter of excentral triangle.

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

X₁₆₄ (INCENTER OF EXCENTRAL TRIANGLE) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Derived Triangle: Excentral Triangle △J_aJ_bJ_c — vertices J_a, J_b, J_c (Anticevian triangle of Incenter X₁ (the three excenters))

• Concurrence Lines: J_a–X₁₆₄, J_b–X₁₆₄, J_c–X₁₆₄ concurring at X₁₆₄

Trilinears: \(\sin B/2 + \sin C/2 - \sin A/2 : \sin C/2 + \sin A/2 - \sin B/2 : \sin A/2 + \sin B/2 - \sin C/2\)

Barycentrics: \(a(\sin B/2 + \sin C/2 - \sin A/2) : b(\sin C/2 + \sin A/2 - \sin B/2) : c(\sin A/2 + \sin B/2 - \sin C/2)\)

Search Value (6, 9, 13): \(5.601337607767\)

▸ Incident Central Lines9 lines

{1, 258}, {3, 3659}, {9, 168}, {40, 188}, {57, 177}, {165, 167}, {173, 504}, {361, 503}, {362, 845}

▸ Functional Relations11 relations

• X₁₆₄ = isogonal conjugate of X₅₀₅

• X₁₆₄ = anticomplement of X₂₁₆₃₃

• X₁₆₄ = X₁-of-excentral triangle

• X₁₆₄ = X₈-of-1st-circumperp-triangle

• X₁₆₄ = X₉₄₄-of-2nd-circumperp-triangle

• X₁₆₄ = excentral-isogonal conjugate of X₁₆₄

• X₁₆₄ = excentral-isotomic conjugate of X₈₄₄

• X₁₆₄ = X₁₈₈-Ceva conjugate of X₁

• X₁₆₄ = X_i-aleph conjugate of X_j for these (i,j): (1, 361), (2, 362), (9, 844), (188, 164), (366, 173)

• X₁₆₄ = centroid of curvatures of extraversions of Conway circle

• X₁₆₄ = ABC-to-excentral trilinear image of X₁

Notes

Let O_A be the circle centered at the A-vertex of the 1st-circumperp-of-1st-circumperp triangle (which is also the 2nd-circumperp-of-2nd-circumperp triangle) and passing through A; define O_B and O_C cyclically. X₁₆₄ is the radical center of O_A, O_B, O_C. (Randy Hutson, August 30, 2020)

Let O_A be the circle centered at the A-vertex of the 1st-circumperp-of-2nd-circumperp triangle (which is also the 2nd-circumperp-of-1st-circumperp triangle) and passing through A; define O_B and O_C cyclically. X₁₆₄ is the radical center of O_A, O_B, O_C. (Randy Hutson, August 30, 2020)