SHARYGIN 2014 Problem 17

Let $AC$ be the hypothenuse of a right-angled triangle $ABC$ . The bisector $BD$ is given, and the midpoints $E$ and $F$ of the arcs $BD$ of the circumcircles of triangles $ADB$ and $CDB$ respectively are marked (the circles are erased). Construct the centers of these circles using only a ruler.