Question 37
Let $ABC$ be a scalene acute angled triangle with $AB \lt AC \lt BC$ and circumcircle $c(O, R)$. The excircle $(c_A)$ corresponding to the vertex $A$ has center $I$ and is tangent to the sides $BC$, $AC$, $AB$ at $D$, $E$, $Z$, respectively. The line $AI$ intersects the circle $c(O, R)$ at $M$ and the circumcircle, say $(c_1)$, of the triangle $AZE$ intersects the circle $(c)$ at $K$. The circumcircle, say $(c_2)$, of the triangle $OKM$ intersects the circle $(c_1)$ at the point $N$. Prove that the lines $AN$ and $KI$ intersect at a point of the circle $(c)$.