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X(1), X(20)


other points below

In general, there is only one point Q whose polar conic in a cubic is a circle and this point Q does not lie on the cubic.

When the cubic is an isogonal pK with pivot P, Q070 is the locus of P such that there is at least one point Q on the cubic having a circular polar conic.

Q070 is a bicircular septic having three real asymptotes which are the altitudes of ABC.

Q070 contains :

– the in/excenters which are nodes on the curve. The nodal tangents are parallel to the asymptotes of the Feuerbach hyperbola for the incenter and the Boutin hyperbolas for the excenters.

– three triple points U, V, W also lying on the Darboux cubic, on K077, on the circle C(O, 3R), on three hyperbolas described in the Darboux cubic page.

– three other points A', B', C' on this same circle and on the homothetic of the McCay cubic under h(O,3). Hence, they are the vertices of an equilateral triangle.

– the de Longchamps point X(20). The corresponding cubic is the Darboux cubic and Q is the circumcenter O. The polar conic is not a proper circle since it decomposes into the line at infinity and the inflexional tangent at O.