Junior Balkan Mathematical Olympiad

Let $O_1$, $O_2$ and $O$ be the circumcenters of triangles $AB{B}^{\prime }$, $AC{C}^{\prime }$ and $ABC$ respectively. As $AB$ is the perpendicular bisector of the line segment $C{C}^{\prime }$, $O_2$ is the intersection of the perpendicular bisector of $AC$ with $AB$. Similarly, $O_1$ is the intersection of the perpendicular bisector of $AB$ with $AC$. It follows that $O$ is the orthocenter of triangle $A{O}_1{O}_2$. This means that $AO$ is perpendicular to $O_1{O}_2$. On the other hand, the segment $A{A}_1$ is the common chord of the two circles, thus it is perpendicular to $O_1{O}_2$. As a result, $A{A}_1$ passes through $O$. Similarly, $B{B}_1$ and $C{C}_1$ pass through $O$, so the three lines are concurrent at $O$.