Question 142
Let $ABC$ be an acute angled triangle with $AB \lt AC \lt BC$, inscribed in the circle $\Gamma_1$ with center $O$. The circle $\Gamma_2$ with center $A$ and radius $AC$ intersects the line $BC$ at point $D$ and the circle $\Gamma_1$ at $E$. The line $AD$ intersects the circle $\Gamma_1$ again at $F$. The circumcircle of the triangle $DEF$ (say, $\Gamma_3$) intersects the line $BC$ at $G$. Prove that:
a) The point $B$ is the center of $\Gamma_3$.
b) The circumcircle of the triangle $CEG$ is tangent to $AC$.
a) The point $B$ is the center of $\Gamma_3$.
b) The circumcircle of the triangle $CEG$ is tangent to $AC$.