Question 190
Given a circle $K$ and a straight line $g$ that have no point in common.
Furthermore, let $\overline{AB}$ be the diameter of $K$, which is orthogonal to $g$, where $B$ is closer to $g$ than $A$. Furthermore, let an arbitrary point $C$, different from $A$ and $B$, be given on $K$. The line $AC$ cuts $g$ into $D$; the line $DE$ touches $K$ in $E$, with $B$ and $E$ lying on the same side of $AC$. Finally, $BE$ intersects the line $g$ at the point $F$ and $AF$ intersects the circle $K$ except in $A$ at the point $G$.
Prove that the mirror point of $G$ lies on the line $CF$ with respect to the axis $AB$.
Furthermore, let $\overline{AB}$ be the diameter of $K$, which is orthogonal to $g$, where $B$ is closer to $g$ than $A$. Furthermore, let an arbitrary point $C$, different from $A$ and $B$, be given on $K$. The line $AC$ cuts $g$ into $D$; the line $DE$ touches $K$ in $E$, with $B$ and $E$ lying on the same side of $AC$. Finally, $BE$ intersects the line $g$ at the point $F$ and $AF$ intersects the circle $K$ except in $A$ at the point $G$.
Prove that the mirror point of $G$ lies on the line $CF$ with respect to the axis $AB$.