I think this is correct....
Based on what we have the area=84 and the semiperimeter=21. Therefore, the incircle(inradius)=84/21=4.
By tangent formulas, we see that the two sides of the quadrilateral are x. So, we have following from the top of the triangle, starting with the side length of 15: x, 15-x, Side length of 14: x-1 and 15-x, Side Length 13: x-1 and x. So, x-1+x=13, 2x-1=13, 2x=14, and x=7. Now, quadrilateral AEIF is made up of two triangles, both with areas of 14, since \(\frac{1}{2}*7*4=14\) . Thus, the answer is \(14+14=\boxed{28}.\)