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K721
equK721

X(101), X(514)

A', B', C' : traces of the antiorthic axis

K721 is an isogonal focal cubic with focus X(101).

The product of distances from M to the sidelines of the reference triangle ABC and the product of distances from M to the sidelines of the excentral triangle IaIbIc are equal if and only if M lies on a sextic which is actually the union of two isogonal nKs with root X(1). One of them is K721, the other is far less interesting (the blue curve on the figure).

When the excentral triangle is replaced with the antimedial triangle, the corresponding sextic splits into K327, the Steiner ellipse and the line at infinity.

When the excentral triangle is replaced with the tangential triangle, the corresponding sextic is the isogonal transform of K327 together with the circumcircle and the line at infinity (Jean-Pierre Ehrmann).

See the analogous cubic K214.

Generalization

Let P be a fixed point. The product of distances from M to the sidelines of the reference triangle ABC and the product of distances from M to the sidelines of the anticevian triangle T(P) of P are equal if and only if M lies on a sextic S(P) passing through A, B, C and bitangent at these points to the sidelines of T(P). S(P) also contains the traces A', B', C' of the trilinear polar L(P) of P and these points are nodes on the curve. See figure below.

K721sextic

When P lies on the Grebe cubic K102, S(P) splits into two (possibly decomposed) cubics nK(#P, P). This is the case of the examples above with P = X(1), X(2), X(6) respectively.

If P* is the isogonal conjugate of P then S(P*) = S(P)*.