Question 49
A triangle $AB\Gamma$ is given, $c(O,R)$ is its circumcircle and let $\Delta$ a point on the side $B\Gamma$ different than the midpoint of $B\Gamma$. The circumcircle of the triangle $BO\Delta$, say $c_1$, intersects the circle $c(O,R)$ at $K$ and the line $AB$ at $Z$. The circumcircle of the triangle $GO\Delta$, say $c_2$, intersects the circle $c(O,R)$ at $M$ and the line $A\Gamma$ at $E$. Finally, the circumcircle of the triangle $AEZ$, say $c_3$, intersects the circle $c(O,R)$ at point $N$. Prove that the triangles $AB\Gamma$ and $KMN$ are equal.