Optimal embedded and enclosing isosceles triangles

Given a triangle $Δ$, we study the problem of determining the smallest enclosing and largest embedded isosceles triangles of $Δ$ 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 $Δ'$ is the smallest area isosceles triangle containing $Δ$, then $Δ'$ and $Δ$ share a side and an angle. In the present paper, we prove that for any triangle $Δ$, every maximum area isosceles triangle embedded in $Δ$ and every maximum perimeter isosceles triangle embedded in $Δ$ shares a side and an angle with $Δ$. Somewhat surprisingly, the case of minimum perimeter enclosing triangles is different: there are infinite families of triangles $Δ$ whose minimum perimeter isosceles containers do not share a side and an angle with $Δ$.

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