Beautiful Geometry 1

Connect the center $B$ of the quarter-circle and the center $E$ of the semi-circle. We have $BE = 5 + 8 = 13$.

Since the black segment $FG$ is tangent to the two circles, it must be orthogonal to $BE$. Since they connect the two pairs of opposite sides of the square, it is easy to see that they are equal.

One way to show this is by constructing a perpendicular $FJ$ from $F$ to $BC$. We have that $FJ = AB = BC$, $∠JGF = 180° - ∠FGC = ∠CEB$, and $∠FJG = ∠BCE = 90°$. Therefore, $△FJG \cong △BCE$, and $FG = BE = 13$.