Olympiad Geometry · Superchallenge problem

Triangle $ABC$ has $AB = 7$, $BC = 8$, and $CA = 9$. The line through $B$ perpendicular to the bisector of $\angle BAC$ meets line $AC$ at $E$. The line through $B$ perpendicular to the bisector of the exterior angle at $A$ meets line $AC$ at $E'$. Find $EC + E'C$.