On transversals of the family of translates of two-dimensional convex compact

The following theorem gives an affirmative answer to Grünbaum's old equistion. Let $\mathscr K$ be the family of translates of a convex compact set $K\subset\mathbb R^2$. If every two elements of $\mathscr K$ have a common point, then there exist three points $A,B,C\in\mathbb R^2$ such that every element of $\mathscr K$ contains some of these points.