Generalizations of Hagge's Theorems

TL;DR

This paper explores two generalizations of Hagge's theorems, extending classical geometric results by replacing the orthocentre with a general point and examining pairs of orthologic triangles, focusing on indirect similarity.

Contribution

It introduces novel generalizations of Hagge's theorems, broadening their applicability to arbitrary points and orthologic triangle pairs, highlighting the role of indirect similarity.

Findings

Generalization involving moving from orthocentre to a general point.

Analysis of pairs of triangles with orthologic centers.

Preservation of indirect similarity in the generalized context.

Abstract

Two generalizations of Hagge's theorems are described. In the first we consider what happens when one moves from the orthocentre to a general point. What one loses by doing so is the indirect similarity and hence one loses the centre of indirect similarity. Instead one proceeds from the centre of the circle under consideration. In the second generalization we consider pairs of triangles that have orthologic centres with respect to each other, so that an indirect similarity is the main feature preserved.