2008 AIME I Problem 10

Let ABCD be an isosceles trapezoid with \overline{AD}||\overline{BC} whose angle at the longer base \overline{AD} is \dfrac{\pi}{3}. The diagonals have length 10\sqrt {21}, and point E is at distances 10\sqrt {7} and 30\sqrt {7} from vertices A and D, respectively. Let F be the foot of the altitude from C to \overline{AD}. The distance EF can be expressed in the form m\sqrt {n}, where m and n are positive integers and n is not divisible by the square of any prime. Find m + n.

Leading zeroes must be inputted, so if your answer is 34, then input 034

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.