Notes
two circles in a square solution

Solution to the Two Circles in a Square Puzzle

Solution by Similar Triangles and Angles in Parallel Lines

There are three similar triangles in the above diagram: $D A E$ and $F B E$ are similar as they are both right-angled triangles that share the angle $F \hat{E} B$. Then $F C D$ is also a right-angled triangle and angle $C \hat{D} F$ is equal to angle $F \hat{E} B$ as they are alternate angles.

The circles give the scale factor from triangle $F D C$ to $D A E$ as $4.5$, meaning that the length of $D A$ is $4.5$ times the length of $C F$.

The area of triangle $F C D$ is half the length of $C F$ times the length of $D C$. The area of the square is the length of $D A$ times the length of $D C$. Therefore the ratio of their areas is the ratio of half the length of $C F$ to the full length of $D A$, which is $1 : 9$. So the fraction that is shaded is $\frac{8}{9}$.