Can any triangle be inscribed in any convex figure? i.e. given a convex figure and a triangle can we transpose and scale and rotate that triangle so that its vertices are on the boundary of the convex figure?

A more general result is known: if $C$ is any Jordan curve and $T$ is a triangle then there exists a triangle similar to $T$ inscribed in $C.$ Moreover, the vertices of such triangles are dense in $C.$ See the references in the Wikipedia article on the Inscribed Square Problem. 

