Question 25
Let $c(O, R)$ be a circle and $A, B$ two points of the circle that are not diametrically opposite. The bisector of the angle $A\hat{B}O$ intersects the circle $c(O, R)$ at point $C$, the circumcircle of the triangle $AOB$ (say $(c_1)$) at point $K$ and the circumcircle of the triangle $AOC$ (say $(c_2)$) at point $L$. Prove that the point $K$ is the circumcenter of the triangle $AOC$ and the point $L$ is the incenter of the triangle $AOB$.