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X(1), X(3), X(15), X(16), X(30)
common points of the circumcircle and the Thomson cubic
Q067 is a bicircular quintic with one real asymptote which is the Euler line.
It is the locus of point P such that the reflection triangle of P is perspective with the symmetric triangle of the isogonal conjugate P* of P. See Table 6.
This is a special case of a more general class of quintics obtained when the symmetric triangle of P* is replaced by the homothetic triangle T(k) of ABC under the homothety with center P*, ratio k. For each k, this triangle is perspective with the reflection triangle of P if and only if P lies on the circumcircle or on a bicircular quintic Q(k) passing through X(1), X(3), X(15), X(16), X(30), the excenters, the common points of the circumcircle and the Thomson cubic.
Obviously, Q067 = Q(2).
With k = 1, we find a classical property of the Neuberg cubic (see property 1) since T(1) is ABC itself.
The limit cases are also interesting :
Q(1/2) contains X(187) and its tangent at O is the Brocard axis.
Q(-1) contains X(2) and its tangent at O is the line through X(74), X(110).
Another generalization is obtained when the reflection triangle of P is replaced by P(k), the homothetic of the pedal triangle of P under the homothety with center P, ratio k. P(k) and T(k) are orthologic and the centers of orthology are P and P*. They are perspective if and only if P lies on a circular quintic Q'(k).
We have Q'(2) = Q067 and Q'(1) is the union of the circumcircle and the Darboux cubic.