KöMaL 2013 B4516

In a triangle ABC , tan $\alpha$=2, tan $\gamma$=1, and $b$=12. The midpoints of the sides opposite to $A$ and $B$ are $F_a$ and $F_b$, and the feet of the corresponding altitudes are $T_a$ and $T_b$, respectively. Prove that the centroid and the orthocentre of triangle ABC are collinear with the intersection of $T_a F_b$ and $F_a T_b$.