Suppose we have given a convex body $K \subset \mathbb{R}^2$. How can we prove that it contains a translate of its homothet $-\frac{1}{2} K$?
hint: take three vertices $A, B$ and $C$ of the convex body and make a triangle $ABC$ inscribed in our convex body and then use the half plans outside the lines $l_A$ (parallel to edge $BC$) , $l_B$ (parallel to edge $AC$) and , $l_C$ (parallel to edge $AB$). (such that triangle $ABC$ be inside the intersection of the half plans ) and then use the homothety mapping with ratio $-\frac{1}{2}$.