you start with 2 pairs of similar triangles here:

DEK and DFG

DJK and DHG

not difficult to prove them similar to each other, right? there are some parallel lines....

Then, you can prove that DE : DF = DJ : DH (= DK : DG). together with common angle D, you can eventually prove the 3rd pair of similar triangles, DEJ and DFH. And since they are similar, angles DEJ = DFH, and finally...