Hey,

In triangle ABC,

F is the midpoint of AB

E is the midpoint of AC

Therefore,

FE//AB & FE=1/2AB(Midpoint theorem).

Now,

In triangle ABP,

F is the midpoint of AB

FQ//BP(FQ and BP are parts of FE and BC respectively)

Therefore,

Q is the midpoint of AP(Converse of midpoint theorem)

Hence, AQ=QP.

Hence, proved.

Hope this helps you!!!!