3. Let $A B C$ be a triangle with incenter $I$. Let the circle centered at $B$ and passing through $I$ intersect side $A B$ at $D$ and let the circle centered at $C$ passing through $I$ intersect side $A C$ at $E$. Suppose $D E$ is the perpendicular bisector of $A I$. What are all possible measures of angle $B A C$ in degrees?
$\angle D I A=\angle D A I=\alpha / 2$. Thus, $\angle I D B=\angle D I B=\alpha$, since $B D=B I$. This gives $\angle D B I=180^{\circ}-2 \alpha$, so that $\angle A B C=360^{\circ}-4 \alpha$. Similarly, $\angle A C B=360^{\circ}-4 \alpha$. Now, summing the angles in $A B C$, we find $720^{\circ}-7 \alpha=180^{\circ}$, so that $\alpha=\frac{540^{\circ}}{7}$.