I have sat and looked at it for a while, but all I have come up with is what I mentioned.

If HE is parallel to HB (in fact runs along it), then HF is parallel to AD, and the ratios are exactly the same. (The two triangles ADB and FHB are *similar* triangles.)

This would also be true if the DE line were aligned along the AH line.

I am hoping another member can take this further, because I have to leave right now, but I will come back later to see how this is going.