Synthetic foundations of cevian geometry II: The center of the cevian conic

TL;DR

This paper explores the properties of a specific conic associated with a triangle and a point P, characterizing its center and linking it to classical triangle centers, using purely synthetic geometric methods.

Contribution

It provides a synthetic geometric characterization of the conic's center related to cevian configurations and introduces a new characterization of the Feuerbach point.

Findings

The conic passes through points related to P and its conjugates.

The center Z is a fixed point of a specific affine map, depending on the conic type.

When P is the Gergonne point, Z is the Feuerbach point.

Abstract

This paper continues the investigation of Part I, by studying the conic $\mathcal{C}_P$ on the five points $ABCPQ$, where $ABC$ is a given ordinary triangle and $Q$ is the isotomcomplement of $P$, defined as the complement of the isotomic conjugate $P'$ of $P$ with respect to triangle $ABC$. We show that $\mathcal{C}_P$ also lies on the points $P'$ and $Q'$, where $Q'$ is the isotomcomplement of $P'$. The conic $\mathcal{C}_P$ lies on six other points which are the images of the vertices of $ABC$ under the affine mapping $\lambda=T_{P'} \circ T_P^{-1}$ and its inverse, where $T_P$ and $T_{P'}$ are the unique affine maps taking $ABC$ to the cevian triangles of $P$ and $P'$, respectively. In the paper we characterize the center $Z$ of $\mathcal{C}_P$ as the unique fixed point of $\lambda$ in the extended plane, when $\mathcal{C}_P$ is a parabola or an ellipse, and the unique ordinary fixed point of $\lambda$, when $\mathcal{C}_P$ is a hyperbola. We also show that $Z=GV \cdot T_P(GV)$, where $G$ is the centroid of $ABC$ and $V=PQ \cdot P'Q'$. When $P$ is the Gergonne point of $ABC$, this gives a new characterization of the Feuerbach point $Z$. All of our arguments are purely synthetic.