DK-GEORGMOHR 2006 Problem 5

We consider an acute triangle $ABC$ . The altitude from $A$ is $AD$ , the altitude from $D$ in triangle $ABD$ is $DE,$ and the altitude from $D$ in triangle $ACD$ is $DF$ . a) Prove that the triangles $ABC$ and $AF E$ are similar. b) Prove that the segment $EF$ and the corresponding segments constructed from the vertices $B$ and $C$ all have the same length.