Linear Families of Triangles and Orthology

Abstract

Two triangles are called orthologic if the perpendiculars from the vertices of one of them to the sides of the other are concurrent. In this paper, we explore the concept of orthology from various points of view. Mostly we work in terms of elementary geometry in $\mathbb{R}^2$. In the final section, we relate the discussed concepts to $\mathbb{R}^4$. A key idea in this paper is that two orthologic triangles generate a one-parameter (linear) family in which any two triangles are orthologic. Working with such a family can provide a natural approach to certain questions.