On Triangles that Contain Triangles

One isn't sure if the following claims are valid. Hope to gain insights on these soon...

1. For any triangle T, the isosceles triangle with minimum area that just contains T necessarily has an angle equal to one of the angles of T.

2. The isosceles triangle with minimum perimeter that just contains T also necessarily has an angle equal to one of the angles of T.

Note: If one looks at the least area/perimeter right triangles that contain a given T:

It *seems* that if T is equilateral, the smallest area right triangle container is isosceles. But for general T, one needs to understand more. Perhaps this problem has already been studied. Shall update this bit soon.

Update-May 25th 2018: Both numbered claims above are not fully valid. They hold only when triangle T is acute. This and a lot more are in a forthcoming paper by Kiss and Pach.