Let $ ABC $ be an acute triangle having $ AB<AC, I $ be its incenter, $ D,E,F $ be intersection of the incircle with $ BC, CA, $ respectively, $ AB, X $ be the middle of the arc $ BAC, $ which is an arc of the circumcicle of it, $ P $ be the projection of $ D $ on $ EF $ and $ Q $ be the projection of $ A $ on $ ID. $ a) Show that $ IX $ and $ PQ $ are parallel. b) If the circle of diameter $ AI $ intersects the circumcircle of $ ABC $ at $ Y\neq A, $ prove that $ XQ $ intersects $ PI $ at $ Y. $
Problem
Source: Stars of Mathematics 2016, Juniors, Problem 4
Tags: geometry, incenter, circumcircle