MEMO 2021 I3

Let $ABC$ be an acute triangle and $D$ an interior point of segment $BC$ . Points $E$ and $F$ lie in the half-plane determined by the line $BC$ containing $A$ such that $DE$ is perpendicular to $BE$ and $DE$ is tangent to the circumcircle of $ACD$ , while $DF$ is perpendicular to $CF$ and $DF$ is tangent to the circumcircle of $ABD$ . Prove that the points $A, D, E$ and $F$ are concyclic.