A Family of Eight-Point Conics Associated with the Cyclic Quadrilateral

We consider the following configuration. Let $ABCD$ be a cyclic quadrilateral with circumcenter $O$, and for each vertex $X$, let $H_X$ be the orthocenter of the triangle formed by the other three. Then $A,B,C,D,H_A,H_B,H_C,H_D$ all lie on a single conic. In this paper we study a certain generalization of this fact as follows. For an arbitrary point $P_D$ on the Euler line of $\mathrm{△}ABC$, we define corresponding points $P_A,P_B,P_C$ on the respective Euler lines such that the ratio $P_X{H}_X:P_{X}O$ is constant for all $X$. We show that the four vertices $A,B,C,D$ and the four isogonal conjugates $Q_A,Q_B,Q_C,Q_D$ of the points $P_X$ all lie on a single conic. This result is given distinct treatments, synthetic, projective, and algebraic. Furthermore, we situate the points $P_X$ within the list of triangle centers.