A property of the orthocentroidal circle

If $ABC$ is a triangle, the circle with $GH$ as diameter, where $G$ and $H$ are the centroid and orthocenter of $ABC$, is called the orthocentroidal circle of $ABC$.

 

We present the following property of the orthocentroidal circle of a triangle: The locus of points $P$ such that the trilinear polar of $P$ goes through the inverse of $P$ on the circumcircle is a quartic, the isogonal conjugate of the orthocentroidal circle.

In the following figure, $Q$ lies on the orthocentroidal circle, $P$ is its isogonal conjugate and $P'$ is the inverse of $P$ on the circumcircle. $A'B'C'$ is the cevian triangle of $P$ and p is the trilinear polar of $P$, through $P'$.