2011 AIME I Problem 2

In rectangle ABCD, AB = 12 and BC = 10. Points E and F lie inside rectangle ABCD so that BE = 9, DF = 8, \overline{BE} \parallel \overline{DF}, \overline{EF} \parallel \overline{AB}, and line BE intersects segment \overline{AD}. The length EF can be expressed in the form m \sqrt{n} - p, where m, n, and p are positive integers and n is not divisible by the square of any prime. Find m + n + p.

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.