VTMO 2013 — Grade 9 — final Question 5

$ABCD$ circumscribes circle $\left(O\right)$, with tangency points $M , N , P , Q$ on $AB , BC , CD , DA$. Let $I = NQ \cap AC$. $a$)Prove $\frac{IA}{IC} = \frac{QA}{NC}$. $b$)Prove $AC , BD , MP , NQ$ concur.