Question 28
On the side $AC$ of the triangle $ABC$ the points $D$ and $E$ are given such that $D$ is between $C$ and $E$. Let $F$ be the intersection of the circumcircle of the triangle $ABD$ and the line through the point $E$ parallel to $BC$ such that $E$ and $F$ are on different sides of the line $AB$. Let $G$ be the intersection of the circumcircle of the triangle $BCD$ and the line through $E$ parallel to $AB$ such that $E$ and $G$ are on different sides of the line $BC$.
Prove that the points $D, E, F$ and $G$ lie on the same circle.
Prove that the points $D, E, F$ and $G$ lie on the same circle.