2016 HMMT November General Round Problem 7

Let ABC be a triangle with AB = 13, BC = 14, CA = 15. The altitude from A intersects BC at D. Let \omega_1 and \omega_2 be the incircles of ABD and ACD, and let the common external tangent of \omega_1 and \omega_2 (other than BC) intersect AD at E. Compute the length of AE.

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 2016 contest; the official solution is available on the HMMT archive. HMMT is not affiliated with or endorsing TopsOJ in any way.