Two Remarkable Ortho-Homological Triangles

TL;DR

This paper explores special triangles called ortho-homological triangles, focusing on two notable examples and their geometric properties, while also generalizing these concepts to polygons and polyhedra in 2D and 3D spaces.

Contribution

It introduces the concept of ortho-homological triangles, highlights two remarkable cases, and extends the theory to polygons and polyhedra in higher dimensions.

Findings

Identification of two remarkable ortho-homological triangles.

Application of Sondat's theorem to these triangles.

Generalization of concepts to polygons and polyhedra.

Abstract

In a previous paper we have introduced the ortho-homological triangles, which are triangles that are orthological and homological simultaneously. In this article we call attention to two remarkable ortho-homological triangles (the given triangle and its first Brocard's triangle), and using the Sondat's theorem relative to orthological triangles, we emphasize on four important collinear points in the geometry of the triangle. Orthological / homological / orthohomological triangles in the 2D-space are generalized to orthological / homological / orthohomological polygons in 2D-space, and even more to orthological / homological / orthohomological triangles, polygons, and polyhedrons in 3D-space.