Notes
five squares solution

Solution to the Five Squares Puzzle

Solution by Similar Triangles

With the points labelled as above, angles $A \hat{C} B$ and $D \hat{C} E$ are the same since they are vertically opposite. Then as triangles $A B C$ and $C D E$ are right-angled and have an angle in common, they are similar.

Let $a$, $b$, and $c$ be the lengths of line segments $C E$, $C D$, and $E D$ respectively. Then the lengths of line segments $A C$, $C B$, and $A B$, are, respectively, $c - b$, $\frac{10 - a}{2}$, and $5$. So by similarity:

The second of these rearranges as follows:

Then using the first:

So $c^2 = 5 \times 10 = 50$. Since $c$ is the side length of the tilted squares, the total shaded area is $100$.

These rearrange to $c(c - b) = 5 a$ and $c(10 - a) = 10 b$.