Perspective Central Triangles Formed from a Triangle and a Transversal

Abstract

Let $\ell$ be a line not passing through any vertex of a triangle $ABC$ and not parallel to any side. Line $\ell$ meets the sidelines $BC$, $CA$, $AB$ of $\triangle ABC$ at points $D$, $E$, $F$, respectively. We consider three of the triangles that are formed: $\triangle AEF$, $\triangle BFD$, and $\triangle CDE$. Placing a fixed triangle center (such as the incenter, centroid, or orthocenter) in each of these three triangles determines a \emph{central triangle}. We investigate when the reference triangle and its central triangle are perspective, i.e., when the lines $AD$, $BE$, and $CF$ are concurrent. A computer search over the first 1000 centers in the Encyclopedia of Triangle Centers suggested numerous examples of concurrence. We give elementary geometric proofs for the circumcenter, orthocenter, and Clawson point, develop a general criterion for concurrence, and identify several operations, including isogonal and isotomic conjugation, that preserve this property. Our main result is a complete characterization of the center functions whose associated cevians are concurrent for every transversal $\ell$. This yields an explicit normal form for such centers. We also show that concurrence depends only on the direction of the transversal, and we investigate the special case in which the transversal is parallel to the Euler line.