VTMO 2026 - Grade 8 - Final - SET 10 - Self-study Question 2

In quadrilateral $ABCD$, points $M$ on $AB$ and $N$ on $CD$ satisfy $BM/BA=DN/DC=1/3$. Construct parallelogram $ABCE$. Let $I,J,O$ be the midpoints of $AC,BD,MN$, respectively. (a) Prove $DE=2IJ$. (b) Prove that $O,I,J$ are collinear.