Triangle A B C has \overline{A B}=5, \overline{B C}=4, \overline{C A}=6. Points D and E are on sides A B and A C, respectively, such that \overline{A D}=\overline{A E}=\overline{B C}. Let C D and B E intersect at F and let A F and D E intersect at G. The length of \overline{F G} can be expressed in the form \frac{a \sqrt{b}}{c} in simplified form. What is a+b+c?