X₄₄₁

COMPLEMENT OF X₂₉₇

X₄₄₁ is the complement of X₂₉₇.

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

X₄₄₁ (COMPLEMENT OF X₂₉₇) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {297, 401}, {2, 3} (Euler line)

• Related Centers: X₂₉₇ (-HIRST INVERSE OF X₄), X₄₀₁ (BAILEY POINT), X₂ (CENTROID), X₃ (CIRCUMCENTER)

Trilinears: \(bc(v + w) : ca(w + u) : ab(u + v)\), where \(u : v : w\) = X₂₄₀; e.g., \(u(A,B,C) = \sec A \cos(A + \omega )\)

Barycentrics: \(v + w : w + u : u + v\)

Search Value (6, 9, 13): \(2.893097558195\)

▸ Incident Central Lines3 lines

{2, 3}, {141, 577}, {525, 647}

▸ Functional Relations9 relations

• X₄₄₁ = midpoint of X₂₉₇ and X₄₀₁

• X₄₄₁ = isotomic conjugate of X₆₃₃₀

• X₄₄₁ = complement of X₂₉₇

• X₄₄₁ = circumcircle-inverse of X₃₇₉₂₁

• X₄₄₁ = crosssum of X₆ and X₂₃₂

• X₄₄₁ = crossdifference of every pair of points on line X₂₅X₆₄₇

• X₄₄₁ = inverse-in-Steiner-inellipse of X₃

• X₄₄₁ = {X₂₄₅₄,X₂₄₅₅}-harmonic conjugate of X₃

• X₄₄₁ = isogonal conjugate of polar conjugate of X₃₀₇₃₇

Notes

As a point on the Euler line, X₄₄₁ has Shinagawa coefficients \((2(E + F)F - S^{2}, S^{2})\)