<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
