The Steiner Point
Let A', B', and C' be the antipodal points of A, B, and C on the Steiner circumconic. The circles AB'C',
BA'C', and CA'B' intersect at the Steiner point, the fourth point of intersection between the ellipse and the circumcircle.
The centers of the circles form a triangle whose centroid is the circumcircle of the original triangle.