PQRS is a parallelogram and G and H are midpoints of QR and RS respectively. PG and PH meet QS at A and B respectively. Prove that QA = AB = BS.

Question

PQRS is a parallelogram and G and H are midpoints of QR and RS respectively. PG and PH meet QS at A and B respectively.

Prove that QA = AB = BS.

Answer 1

Let $PR\cap QS=\{O\}$.

Thus, $QO$ and $PG$ are medians of $\Delta PQR$.

Thus, $$QA:AO=2:1.$$ Similarly, $$SB:BO=2:1$$ and since $QO=OS,$ we are done!

For example. Let $AO-x$ and $BO=y$.

Hence, $$3x=QO=OS=3y,$$ which says $x=y$ and $QA=AB=BS.$