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K662

a (b + c) x (c^2y^2 + b^2z^2) + 2 abc(a+b+c) xyz = 0

X(100), X(513)

A', B', C' : traces of the trilinear polar of X(37)

K662 is the locus of P such that the sum of the six algebraic distances from P and its isogonal conjugate Q to the sidelines of ABC (sometimes called absolute normal coordinates) is zero. See the related cubic K661.

K662 is an isogonal focal nK, the locus of the foci of inscribed conics with center on the antiorthic axis or equivalently tangent to the trilinear polar of X(37).

Its singular focus is X(100), the isogonal conjugate of X(513) = infinite point of the antiorthic axis.

K662a

Simultaneous construction of K661 and K662

A variable line (L) parallel to the antiorthic axis (A) is reflected about (A) into (L') and is transformed into the circum-conic (C) passing through X(100) under isogonal conjugation.

(C) meets (L) at M1, M2 on K661 and (L') at N1, N2 on K662.

If P* is replaced with the image of P under the isoconjugation with pole Ω, then the locus of P becomes nK(Ω, X37, Ω÷X513) where Ω÷X513 is the Ω–isoconjugate of X513.

This cubic is a nK0 if and only if Ω lies on the antiorthic axis.

It is a central cubic nK++ if and only if Ω lies on the trilinear polar of X(649) and consequently a nK0++ when Ω = X(3768).

It is a nodal cubic cK if and only if Ω lies on the Steiner inscribed ellipse or on the Brocard ellipse.