### Instruction: Square $ABCD$ has area $200$. Point $E$ lies on side $\overline{BC}$. Points $F$ and $G$ are the midpoints of $\overline{AE}$ and $\overline{DE}$, respectively. Given that quadrilateral $BEGF$ has area $34$, what is the area of triangle $GCD$?### Response: To start, I notice that the problem involves a lot of areas and shapes, so I think it would be helpful to draw a diagram.
I sketch a square $ABCD$ and label the vertices and the area.
Then I mark point $E$ on $\overline{BC}$ and draw $\overline{AE}$ and $\overline{DE}$.
Next, I find the midpoints $F$ and $G$ of $\overline{AE}$ and $\overline{DE}$, respectively, and connect them to form quadrilateral $BEGF$.
I label the area of $BEGF$ as $34$.
Finally, I shade triangle $GCD$ to indicate that I want to find its area.
Here is what my diagram looks like: