KöMaL 2014 C1254

In a triangle $ABC$ , $T$ is the foot of the altitude drawn from $C$ , and $AT=3BT$ . Let $F$ denote the midpoint of $AB$ , and let $D$ denote the point of altitude $CT$ where $AB$ subtends a right angle. Prove that if the orthocentre of triangle $ABC$ coincides with the centroid of triangle $FBD$ then $AD$ bisects the angle $BAC$ .