jmc

We begin by drawing a diagram: We know that the gray area above (quadrilateral $BEGF$) has area $34$, and we wish to determine the pink area ($△GCD$).

First we note that $△AED$ has base $AD$, equal to the side length of square $ABCD$, and also has height equal to the side length of square $ABCD$. Thus $△AED$ has area equal to half the area of $ABCD$, or $100$.

Triangle $△FEG$ has half the base and half the height of $△AED$, so its area is $\frac{1}{2}\cdot \frac{1}{2}\cdot 100=25$.

Since quadrilateral $BEGF$ can be divided into $△FEG$ and $△FBE$, we know that $△FBE$ has area $34-25=9$. This is half the area of $△ABE$ (which shares an altitude with $△FBE$ and has twice the corresponding base). Thus, $△ABE$ has area $18$.

Since square $ABCD$ can be divided into triangles $ABE$, $AED$, and $ECD$, we know that the area of $△ECD$ is $200-100-18=82$. Finally, $△GCD$ shares an altitude with $△ECD$ and has half the corresponding base, so the area of $△GCD$ is $\frac{1}{2}\cdot 82$, or $\boxed{41}$.