Let $\triangle{ABC}$ have side lengths $\overline{AB}=3$, $\overline{BC}=5$, $\overline{CA}=7$ and let $D$ be the circumcenter of this triangle. Also, let $E$ be the incenter of $\triangle{ABD}$. $F$ is the incenter of $\triangle{ACD}$. Find the length of segment $\overline{EF}$.
This problem is begging for a diagram, but herein lies the problem. It is difficult to construct a $3-5-7$ triangle. Even more difficult to find the circumcenter and incenter of these triangles. Help is much appreciated and needed.
HINT
If this is the main problem then here is how to construct a 3-5-7 triangle. Assume that the radius of the little white circles is one. Then