Then $D, E, F, G, H, I$ lie on a same circle or no?

Question

In the figure If $GHFI, DFEG, GHEF$ are concyclic. Then $D, E, F, G, H, I$ lie on a same circle or no?

Answer 1

Given any $3$ non-collinear points, there exists one and only one circle passing through all of them. So the circle $DFEG$ (call it $\Gamma$) is the only circle that passes through $E$, $F$ and $G$. But $GHEF$ is also cyclic. So $H$ is also a point on $\Gamma$. $\Gamma$ is the only circle that passes through $F$, $G$ and $H$. But $GHFI$ is a cyclic quadrilateral. So $I$ also lies on $\Gamma$. All of $D$, $E$, $F$, $G$, $H$ and $I$ lies on the circle $\Gamma$.