The Anticevian Intersection Conic

Given a triangle $ABC$ and a point $P$, we consider the cevian and anticevian triangles of $P$, $A'B'C'$ and $A''B''C''$, we consider six intersection points, all lying on the same conic. Which type of conic is this according to the position of $P$? When is this conic a circle?

The Anticevian Intersection Conic