<Tri-2026-2380>, hint

Suppose line DI  cuts incircle at H and AG at K respectively,

and AI cut EF at M, midpoint of EF and so  <AMG=90 

Suppose a circle AGM (AG is diameter) cut incircle at J. <AJG=90=<AJD

Suppose JA cut incircle  at H'. Since <H'JD=90, H' lie on line DI, H' lie on incircle, and H'(=H) is the intersection of AJ and DK, which is orthocenter of triangle ADG