VTMO 2016 — Grade 8 — final Question 6

In quadrilateral $ABCD , M$ on $AB , N$ on $CD$ satisfy $\frac{BM}{BA} = \frac{DN}{DC} = \frac{1}{3}$. Construct parallelogram $ABCE$. Let $I , J , O$ be midpoints of $AC , BD , MN$. Prove $a ) DE = 2 IJ$; $b ) O , I , J$ are collinear.