VTMO 2026 - Grade 8 - Semifinal - SET 09 - Self-study Question 10

Triangle $ABC$ has $AB=5$, $AC=7$, $BC=8$. The internal angle bisectors $BE,CF$ meet at $I$. Points $M,N$ lie on $BC$ with $BM=5$, $CN=7$. Lines $ME,NF$ meet at $K$, and $IK$ meets line $AB$ at $D$.
(a) Prove $\angle ABC=60^\circ$.
(b) Find $AD$.