Generalization of the Wallace-Simson line

TL;DR

This paper explores a generalization of the Wallace-Simson line by allowing the midpoint to move along a specific locus, revealing connections to known geometric configurations.

Contribution

It introduces a generalized form of the Wallace-Simson line by varying the midpoint along the perpendicular bisector, linking it to existing geometric concepts.

Findings

Generalized Wallace-Simson lines are linked to known geometric configurations.

The midpoint Q can move along the perpendicular bisector of HJ.

The generalization reveals new insights into classical triangle geometry.

Abstract

A Wallace-Simson line of a point J has the property that it passes through the midpoint Q of HJ, where H is the orthocentre. By allowing Q to move on the perpendicular bisector of HJ we obtain generalized Wallace-Simson lines. These turn out to be known in another context.