14. (BUL 5) A line in the plane of a triangle $A B C$ intersects the sides $A B$ and $A C$ respectively at points $X$ and $Y$ such that $B X=C Y$. Find the locus of the center of the circumcircle of triangle $X A Y$.
14. Let $O, D, E$ be the circumcenter of $\triangle A B C$ and the midpoints of $A B$ and $A C$, and given arbitrary $X \in A B$ and $Y \in A C$ such that $B X=C Y$, let $O_{1}, D_{1}, E_{1}$ be the circumcenter of $\triangle A X Y$ and the midpoints of $A X$ and $A Y$, respectively. Since $A D=A B / 2$ and $A D_{1}=A X / 2$, it follows that $D D_{1}=B X / 2$ and similarly $E E_{1}=C Y / 2$. Hence $O_{1}$ is at the same distance $B X / 2=C Y / 2$ from the lines $O D$ and $O E$ and lies on the halfline bisector $l$ of $\angle D O E$.
If we let $X, Y$ vary along the segments $A B$ and $A C$, we obtain that the locus of $O_{1}$ is the segment $O P$, where $P \in l$ is a point at distance $\min (A B, A C) / 2$ from $O D$ and $O E$.