Synthetic foundations of cevian geometry, IV: the TCC-perspector theorem

TL;DR

This paper provides a synthetic proof of the TCC-perspector theorem, relating the isogonal conjugate of the generalized orthocenter to perspectors of specific triangles in cevian geometry.

Contribution

It offers a new synthetic proof of the TCC-perspector theorem, connecting various cevian and conjugate points in triangle geometry.

Findings

Synthetic proof of the TCC-perspector theorem

Identification of the perspector between the tangential and circumcevian triangles

Clarification of relationships among isogonal, isotomic, and complement points

Abstract

In this paper we give a completely synthetic proof of the TCC-perspector theorem, that the isogonal conjugate $\gamma(H)$ of the generalized orthocenter $H$ (defined in Part III of this series of papers), with respect to a triangle $ABC$ and a point $P$, is the perspector of the tangential triangle of $ABC$ and the circumcevian triangle (both with respect to the circumcircle) of the isogonal conjugate $\gamma(Q)$, where $Q$ is the complement of the isotomic conjugate $P'$ of the point $P$.