KöMaL 2009 B4200

For a finite point set $A$ in the plane, let $v(A)$ denote the number of triangles in a triangulation of set $A$ , and let $A+A=\{x+y\mid x,y\in A\}$, where the sum of two points is defined as the point whose position vector is the sum of the position vectors of the terms. Prove that $v(A+A)\ge 4v(A)$.