Synthetic foundations of cevian geometry, I: Fixed points of affine maps in triangle geometry

Abstract

We give synthetic proofs of many new results in triangle geometry, focusing especially on fixed points of certain affine maps which are defined in terms of the cevian triangle $DEF$ of a point $P$ with respect to a given triangle $ABC$, as well as the cevian triangle of the isotomic conjugate $P'$ of $P$ with respect to $ABC$. We prove a formula for the cyclocevian map in terms of the isotomic and isogonal maps using an entirely synthetic argument, and show that the complement $Q$ of the isotomic conjugate $P'$ has many interesting properties. If $T_P$ is the affine map taking $ABC$ to $DEF$, we show synthetically that $Q$ is the unique ordinary fixed point of $T_P$ when $P$ is any point not lying on the sides of triangle $ABC$, its anti-complementary triangle, or the Steiner circumellipse of $ABC$. We also show that $T_P(Q')=P$ if $Q'$ is the complement of $P$, and that the affine map $T_P T_{P'}$ is either a homothety or a translation which always has the $P$-ceva conjugate of $Q$ as a fixed point. Finally, we show that $P$ lies on the Steiner circumellipse if and only if $T_PT_{P'}=K^{-1}$, where $K$ is the complement map for $ABC$. This paper forms the foundation for several more papers to follow, in which the conic on the 5 points $A,B,C,P,Q$ is studied and its center is characterized as a fixed point of the map $\lambda=T_{P'} T_P^{-1}$.