An Isosceles Triangle Theorem

ABE is an Isosceles triangle.  D lies on AB and C lies on BE extended so that AD = CE.  We illustrate the fact that DF = FC:

An interesting question is this:  what is the ratio of the lengths AF:FE in terms of BD and DA.  We can investigate this by setting a couple of values for these quantities and inspecting the output.

First we set BD to 3, and DA to 2:

Now we inspect AF and FE and examine their ratio:

3:2 input ratio leads to 7/3 output ratio.  How about a 3:1 input ratio:

Can you come up with a formula?  Try some more different inputs.