Minimum area isosceles containers

We show that every minimum area isosceles triangle containing a given triangle $T$ shares a side and an angle with $T$. This proves a conjecture of Nandakumar motivated by a computational problem. We use our result to deduce that for every triangle $T$, (1) there are at most $3$ minimum area isosceles triangles that contain $T$, and (2) there exists an isosceles triangle containing $T$ whose area is smaller than $\sqrt{2}$ times the area of $T$. Both bounds are best possible.