X₄₈₈

ANTICOMPLEMENT OF X₄₈₅

X₄₈₈ is the anticomplement of X₄₈₅.

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

X₄₈₈ (ANTICOMPLEMENT OF X₄₈₅) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {485, 641}

• Related Centers: X₄₈₅ (VECTEN POINT), X₆₄₁ (COMPLEMENT OF X₄₈₅)

Trilinears: \(f(A,B,C) : f(B,C,A) : f(C,A,B)\), where \(f(A,B,C) = (\cos A)(1 + \csc A \sin B \sin C)\)

Barycentrics: \((b^{2} + c^{2} - a^{2})(a^{2} + 2\sigma ) : (c^{2} + a^{2} - b^{2})(b^{2} + 2\sigma ) : :\)

Search Value (6, 9, 13): \(6.252816678238\)

▸ Incident Central Lines7 lines

{2, 372}, {3, 69}, {4, 490}, {193, 371}, {376, 489}, {491, 631}, {591, 3071}

▸ Functional Relations7 relations

• X₄₈₈ = reflection of X₄₈₅ in X₆₄₁

• X₄₈₈ = isogonal conjugate of X₈₉₄₈

• X₄₈₈ = anticomplement of X₄₈₅

• X₄₈₈ = anticomplementary conjugate of X₆₃₇

• X₄₈₈ = X_i-Ceva conjugate of X_j , for these (i,j): (4, 487), (490, 20), (492, 2)

• X₄₈₈ = X₂₀-of-outer-Vecten-triangle

• X₄₈₈ = orthologic center of inner Vecten to outer Vecten triangles

Notes

X₄₈₈ is a perspector of triangles associated with squares that circumscribe ABC. For details, see Vierkanten in een driehoek: 2. Meer punten uit omgeschreven vierkanten (Floor van Lamoen, 4/29/98)

Let O_A be the circle centered at the A-vertex of the outer Vecten triangle and passing through A; define O_B and O_C cyclically. X₄₈₈ is the radical center of O_A, O_B, O_C. (Randy Hutson, August 30, 2020)