2014 AIME I Problem 15

In \triangle ABC, AB = 3, BC = 4, and CA = 5. Circle \omega intersects \overline{AB} at E and B, \overline{BC} at B and D, and \overline{AC} at F and G. Given that EF=DF and \frac{DG}{EG} = \frac{3}{4}, length DE=\frac{a\sqrt{b}}{c}, where a and c are relatively prime positive integers, and b is a positive integer not divisible by the square of any prime. Find a+b+c.

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.