MathNet 2020 — Math Olympiad Vault — Part 05 Question 121

The points $M \in (AB)$, $N \in (BC)$ and $P \in (CD)$ are chosen on three sides of the rhombus $ABCD$. Prove that the centroid of the triangle $MNP$ belongs to the line $AC$ if and only if $AM + DP = BN$.