<Qua-2026-822>, hint

 

<Qua-2026-822>, rectangle, others

<South Africa MO 2017> 

<Tri-2026-2333>, hint

AFIE are concyclic and  since <EPF=<A, A,P,F,I,E are concyclic too.

<PFE=<C=<PIE. <EID=180-<C. Hence <EID+<PIE=180.

Hence P I D are collinear. 

<Tri-2026-2333>, general, incenter

<South Africa MO 2016>

 

<Tri-2026-2332 >, hint