Question 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 $a + b + c.$