Location of incenters and Fermat points in variable triangles

TL;DR

This paper investigates the positions of incenters and Fermat points within the orthocentroidal circle of triangles, revealing they lie inside this circle and exploring their relation to the Euler line.

Contribution

It demonstrates that incenters and Fermat points of triangles with a fixed Euler line are contained within the orthocentroidal circle, providing new geometric insights.

Findings

Incenters of such triangles cover the interior of the orthocentroidal circle.

Fermat points also lie within this circle.

The orthocentroidal circle has diameter GH, connecting the centroid and orthocenter.

Abstract

The orthocentroidal circle of a nonequilateral triangle has diameter GH, joining the centroid to the orthocenter. We show that the incenters of triangles with a given Euler line simply cover the interior of the orthocentroidal circle, and that their Fermat points also lie within this circle.