jmc

Since $E$ is the midpoint of $\overline{BC}$, it has coordinates $(\frac{1}{2}(8+0),\frac{1}{2}(0+0))=(4,0)$. The line passing through the points $A$ and $E$ has slope $\frac{6-0}{0-4}=-\frac{3}{2}$; the $y$-intercept of this line is the $y$-coordinate of point $A$, or 6. Therefore, the equation of the line passing through points $A$ and $E$ is $y=-\frac{3}{2}x+6$. Point $F$ is the intersection point of the lines with equation $y=-\frac{3}{8}x+3$ and $y=-\frac{3}{2}x+6$. To find the coordinates of point $F$ we solve the system of equations by equating $y$: $$\begin{array}{rl}-\frac{3}{8}x+3& =-\frac{3}{2}x+6\\ 8(-\frac{3}{8}x+3)& =8(-\frac{3}{2}x+6)\\ -3x+24& =-12x+48\\ 9x& =24\end{array}$$Thus the $x$-coordinate of point $F$ is $x=\frac{8}{3}$; it follows that $y=-\frac{3}{2}\times \frac{8}{3}+6=2$. Hence $F=(\frac{8}{3},2)$ and sum of its coordinates are $\frac{8}{3}+2=\frac{8}{3}+\frac{6}{3}=\boxed{\frac{14}{3}}$.