2014 AIME I Problems/Problem 15

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.