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

TL;DR

The paper extends the classical eight-point conic phenomenon for a cyclic quadrilateral by introducing a parametric family of points on the Euler lines, linked through a common ratio to their Euler-line centers: for each vertex $X$, the associated $P_X$ satisfies $\frac{P_X H_X}{P_X O}$ equalized across the four Euler lines, and $Q_X$ is the isogonal conjugate of $P_X$. It proves that the eight points $A,B,C,D,Q_A,Q_B,Q_C,Q_D$ lie on a single conic, with proofs offered in synthetic, projective, and algebraic styles, and it connects the construction to triangle-center theory via Shinagawa coefficients. The work further analyzes centers with constant Shinagawa coefficients, providing a catalog of candidates and an explicit equation $\Phi(u,v)$ for the eight-point conic family in barycentric coordinates, thereby unifying cyclic-quadrilateral geometry, Euler-line dynamics, and ETC centers. This framework enhances the understanding of how isogonal conjugation and conic loci interact in cyclic configurations and offers a concrete route to identifying centers that generate the same conic family.

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.

Figures (5)

• Figure 1: $A, B, C, D, Q_{A}, Q_{B}, Q_{C}, Q_D$ lie on a conic.
• Figure 2: Point $T$ lies on the perpendicular bisector of the segment $BC$.
• Figure 3: $TT_3$ goes through the point $\infty_D$.
• Figure 4: $M_{BC}$ is the center of the hyperbolas.
• Figure 5: $k$ and $\ell$ are reflections of each other in the perpendicular bisector of $BC$.

Theorems & Definitions (28)

• Proposition 1.1
• Lemma 2.1: Steiner's Theorem
• Lemma 2.2: Isogonal Conjugate of a Line
• Remark 2.3
• Lemma 2.4
• proof
• Theorem 2.5
• Remark 2.6: Degenerated cases
• proof
• Lemma 2.7