Notes
three squares in a square solution

Three Squares in a Square

Solution by Similar Triangles

With the points labelled as above, let $a$ be the length of a side of a smaller square and let $b$ be the distance between the squares. These are then two of the sides of triangle $E D C$. Triangle $C B A$ then has shorter sides of lengths $a$ and $a + b$. Since these two triangles are similar, the ratios $a : b$ and $a + b : a$ are equal, meaning that $a b + b^2 = a^2 = 5$.

The shaded region can be viewed as the square $H F B A$ without the two triangles $H G A$ and $C B A$. The combined area of these triangles is $a (a + b)$ meaning that the shaded region has area $b(a + b) = 5$.