Optimal embedded and enclosing isosceles triangles

TL;DR

This paper investigates the properties of the largest embedded and smallest enclosing isosceles triangles within any given triangle, revealing shared sides and angles in most cases but also identifying exceptions for perimeter minimization.

Contribution

It proves that maximum area and perimeter isosceles triangles embedded in a triangle share a side and an angle with it, extending previous results and highlighting new exceptions.

Findings

Maximum area isosceles triangles share a side and an angle with the original triangle.

Maximum perimeter isosceles triangles also share a side and an angle with the original triangle.

There are infinite cases where minimum perimeter enclosing isosceles triangles do not share a side and an angle.

Abstract

Given a triangle $\Delta$, we study the problem of determining the smallest enclosing and largest embedded isosceles triangles of $\Delta$ with respect to area and perimeter. This problem was initially posed by Nandakumar and was first studied by Kiss, Pach, and Somlai, who showed that if $\Delta'$ is the smallest area isosceles triangle containing $\Delta$, then $\Delta'$ and $\Delta$ share a side and an angle. In the present paper, we prove that for any triangle $\Delta$, every maximum area isosceles triangle embedded in $\Delta$ and every maximum perimeter isosceles triangle embedded in $\Delta$ shares a side and an angle with $\Delta$. Somewhat surprisingly, the case of minimum perimeter enclosing triangles is different: there are infinite families of triangles $\Delta$ whose minimum perimeter isosceles containers do not share a side and an angle with $\Delta$.