Quadrilateral $ABCD$ is a rectangle. The points $E$ and $G$ are the midpoints of $DC$ and $BC$ respectively. The point $F$ is the intersection of the diagonal $AC$ and $BE$. Prove that $D, F,$ and $G$ are collinear.![enter image description here](https://i.stack.imgur.com/Oih8f.png)

I have included a sketch of the problem. You can see where I constructed an extension of $AB$ and a line through the points $DG$ which intersects $AB$ at $I$. I can show that $BGI$ and $DGC$ are congruent triangles by $ASA$, but I am not sure how to show that $D, F,$ and $G$ are collinear.