<Tri-2024-2113>, hint

Since <DIP=1/2(B+C), <IDP=<A, hence <CDP=<B/2, and so DP//AE (<EAC=<B/2=<CQP

Hence D,P,C,O are concyclic,  <EQD=<C/2, <AQE=<C=<ACE

DQ is angled bisector of <AQE and <DAE=<DEA=<DBE=<B/2

Hence QD is perpendicular to AE and DP (AE//DP)