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

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 $P^{\mathrm{\prime }}$ of $P$ with respect to triangle $ABC$. We show that ${\mathcal{C}}_P$ also lies on the points $P^{\mathrm{\prime }}$ and $Q^{\mathrm{\prime }}$, where $Q^{\mathrm{\prime }}$ is the isotomcomplement of $P^{\mathrm{\prime }}$. The conic ${\mathcal{C}}_P$ lies on six other points which are the images of the vertices of $ABC$ under the affine mapping $\lambda =T_{P^{\mathrm{\prime }}}\circ T_P^{-1}$ and its inverse, where $T_P$ and $T_{P^{\mathrm{\prime }}}$ are the unique affine maps taking $ABC$ to the cevian triangles of $P$ and $P^{\mathrm{\prime }}$, 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 $V=PQ\cdot P^{\mathrm{\prime }}{Q}^{\mathrm{\prime }}$. 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.