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infinite points of the sidelines of ABC

K327 is a Tucker cubic, the locus of M whose cevian triangle has area the double of that of ABC. See "Tucker cubics" in the Downloads page. See also the other Tucker cubics K011 up to K016. The cubics K015, K016 and K327 are three members of CL064.

It is a nK(X2, X2, ?) without any known center on it. The three real inflexional asymptotes are the sidelines of the medial triangle. The tangents at A, B, C are the sidelines of the antimedial triangle. Hence, it is tritangent to the Steiner ellipse at these points.

The isogonal transform of K327 is also a member of CL064.

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 antimedial triangle are equal if and only if M lies on a sextic which is actually the union of K327, the Steiner ellipse and the line at infinity. See the analogous K721.

In his book "Propriétés projectives", t.II, p.384, Poncelet claimed that the number of (real) tangents one can draw to a curve of order (degree) m was at most m(m-1). Gergonne argued that he couldn't see a cubic such that six (real) tangents could be drawn from a suitably chosen point. K327 gives a simple example of such cubic. Indeed, it is possible to draw six real tangents from the centroid G to the curve and their contacts lie on an ellipse homothetic of the Steiner ellipse under the homothety with center G, ratio sqrt(7). This ellipse also contains the six points symmetric of a vertex of ABC in the two others.

K327 is the jacobian of three ellipses Ea, Eb, Ec. Ea has center A, passes through B, C, the reflections of B and C in A such that AB, AC are two conjugated diameters. It is tangent at B, C to the sidelines of the antimedial triangle and to the Steiner ellipse. Ea has equation : x^2+2(yz+zx+xy)=0.

This means that K327 is the locus of M such that the three polars La, Lb, Lc of M in Ea, Eb, Ec respectively concur at N which is also a point on the cubic.

(quoted from Brocard & Lemoyne, Courbes géométriques remarquables, t. III, p. 84 with corrections and complements)

See here for another property (Angel Montesdeoca, in Spanish, 2022-02-03)