Question 111
Let $ABC$ be an acute angled triangle, $AA_1$ and $CC_1$ be its bisector, $I$ be the incenter of $ABC$, $M$ and $N$ be the midpoints of $AI$ and $CI$, respectively. Inside the triangles $AC_1I$ and $A_1CI$ we choose points $K$ and $L$, such that
$\angle AKI = \angle CLI = \angle AIC$, $\angle AKM = \angle ICA$, $\angle CLN = \angle IAC$. Prove that radiuses of the circumcircles of the triangles $KIL$ and $ABC$ are equal.
$\angle AKI = \angle CLI = \angle AIC$, $\angle AKM = \angle ICA$, $\angle CLN = \angle IAC$. Prove that radiuses of the circumcircles of the triangles $KIL$ and $ABC$ are equal.