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 √2 times the area of T. Both bounds are best possible.