The incenter of a triangle as a cone isoperimetric center

TL;DR

This paper demonstrates that the incenter of a triangle can be characterized as the isoperimetric center of a cone constructed over the triangle, linking geometric optimization with classical triangle centers.

Contribution

It introduces a novel geometric interpretation of the incenter as an isoperimetric center of a cone over the triangle, connecting cone optimization with triangle center theory.

Findings

The incenter coincides with the cone's isoperimetric center.

The minimal ratio of boundary area cubed to volume squared defines the incenter.

A regular projection of a cone vertex over a triangle yields this incenter.

Abstract

We show that the the image of the regular projection of a vertex of a cone over a triangle that minimizes the ratio of the cube of the area of the boundary of the cone and the square of the volume of the cone coincides with the incenter.