  We study the problem of finding maximum-area triangles that can be inscribed
in a polygon in the plane. We consider eight versions of the problem: we use
either convex polygons or simple polygons as the container; we require the
triangles to have either one corner with a fixed angle or all three corners
with fixed angles; we either allow reorienting the triangle or require its
orientation to be fixed. We present exact algorithms for all versions of the
problem. In the case with reorientations for convex polygons with $n$ vertices,
we also present $(1-\varepsilon)$-approximation algorithms.
