Notes
two rectangles with diagonals solution

Solution to the Two Rectangles with Diagonals Puzzle

Solution by Similar Triangles and Equilateral Triangles

First, note that the rectangles are similar since the angle between a side and the diagonal is the same for each (angle $H \hat{B} G$). So the ratio of the side to the half-diagonal is the same for each, so the ratio of the lengths of $B G$ to $B H$ is the same as with them the other way round, which means that they must be equal in length. This is also the length of $G H$ so triangle $B G H$ is equilateral and quadrilateral $E G B H$ is a kite.

Since triangle $B G H$ is equilateral, triangle $G I B$ is half an equilateral triangle of the same side length. Each rectangle therefore has area four times that of the triangle $B G H$.

Triangle $J G H$ is then half of an equilateral triangle, so $H G$ has double the length of $G J$. The point $K$ is the midpoint of $G H$ so $H K$ is the same length as $J G$ and triangles $G J E$ and $E K H$ are congruent. Triangle $E G H$ is therefore two thirds of $G J H$ and so its area is one third of that of $B G H$. The area of the purple region is therefore $\frac{4}{3}$rds of the area of $B G H$.

The total area is then $8 - \frac{4}{3} = \frac{20}{3}$ times the area of $B G H$, so the fraction that is shaded is $\frac{4}{20} = \frac{1}{5}$th of the total area.