  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
$\Delta$.
