Notes
square with midpoints solution

Solution to the Square with Midpoints Puzzle

Solution by Similar Triangles

In the above diagram, the point labelled $G$ is directly below $E$.

Triangle $A E G$ is similar to triangle $A D C$, while triangle $G B E$ is similar to $A B F$. So the length of $G B$ is half of $E G$ which is half of $A G$. So $G B$ is one fifth of $A B$, and so $E G$ is two fifths of $A B$ and so a fifth of $A C$. This means that triangle $A E B$ is one twentieth of the area of the full square. Triangles $A D C$ and $A B F$ are both one quarter of the area of the square. Therefore the shaded area is the following fraction of the square: