MathNet 2022-part-01 — 197 problems (EN–VI) Question 61

Point $D$ is marked on the half-circle with the diameter $AB$ and the center $O$. The points $E$ and $F$ are the midpoints of lesser arcs $AD$ and $BD$ respectively. It is known that $O$ lies on the line passing through the orthocenters of the triangles $ADF$ and $BDE$.
Find all possible values of the angle $AOD$.