RO-NMO 2005 Problem 91

Let $ABCD$ be a convex quadrilateral with $AD\not\parallel BC$ . Define the points $E=AD \cap BC$ and $I = AC\cap BD$ . Prove that the triangles $EDC$ and $IAB$ have the same centroid if and only if $AB \parallel CD$ and $IC^{2}= IA \cdot AC$ .

Virgil Nicula