Problem 3. Let ABC be an acute triangle. Points B, D, E, and C lie on a line in this order and satisfy BD = DE = EC. Let M and N be the midpoints of AD and AE, respectively. Suppose triangle ADE is acute, and let H be its orthocentre. Points P and Q lie on lines BM and CN , respectively, such that D, H, M , and P are concyclic and pairwise different, and E, H, N , and Q are concyclic and pairwise different. Prove that P , Q, N , and M are concyclic.
The orthocentre of a triangle is the point of intersection of its altitudes.