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

TL;DR

This paper explores a family of eight-point conics related to cyclic quadrilaterals and their generalizations, establishing new geometric configurations involving Euler lines, isogonal conjugates, and triangle centers through synthetic, projective, and algebraic methods.

Contribution

It introduces a generalized framework for eight-point conics associated with cyclic quadrilaterals, linking them to Euler lines and triangle centers with multiple proof techniques.

Findings

All eight points lie on a single conic in the generalized setting.

The points $P_X$ are related to known triangle centers.

The configuration holds under various geometric perspectives.

Abstract

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 $\triangle ABC$, we define corresponding points $P_A, P_B, P_C$ on the respective Euler lines such that the ratio $P_XH_X : P_XO$ 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.