Journal for Geometry and Graphics 29 (2025), No. 2, 173--185
Copyright by the authors licensed under CC BY SA 4.0
A Family of Eight-Point Conics Associated with the Cyclic Quadrilateral
Kazimierz Chomicz
Jagiellonian University, Cracow, Poland
kazikchomicz@gmail.com
Milosz Platek
Cracow, Poland
milosz@platek.org
Konstanty Smolira
Jagiellonian University, Cracow, Poland
konstanty.smolira@student.uj.edu.pl
Dylan Wyrzykowski
University of Warsaw, Warsaw, Poland
d.wyrzykows2@student.uw.edu.pl
[Abstract-pdf]
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 $P_X H_X : P_X O$ 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.
Keywords: Euler line, isogonal conjugate, conic, cyclic quadrilateral, triangle center, orthocenter, Shinagawa coefficients.
MSC: 51M04; 51M05, 51M15.
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