2013 HMMT November General Round Problem 5

Let ABC be an isosceles triangle with AB = AC. Let D and E be the midpoints of segments AB and AC, respectively. Suppose that there exists a point F on ray \overrightarrow{DE} outside of ABC such that triangle BFA is similar to triangle ABC. Compute \frac{AB}{BC}.

Answers are checked by value, so any equivalent form is accepted: 1/2, \frac{1}{2} and 0.5 all count as the same answer.

Full credit goes to HMMT for authoring these problems. This problem is from the November 2013 contest; the official solution is available on the HMMT archive. HMMT is not affiliated with or endorsing TopsOJ in any way.