Take the convex hull of the points, let it be a $k$-gon. If $k=2015$, done. If $k<2015$, there is a point $P$ inside the convex hull. Hence, $P$ is in some triangle $ABC$ formed by vertices of the convex hull (because we can triangulate the $k$-gon). Then $A,B,C,P$ give a contradiction.