Notes
two rectangles in a square solution

Solution to the Two Rectangles in a Square Puzzle

Solution by Area

As the dots are equally spaced, joining opposite dots divides the outer square into nine squares of equal size. The red corner triangles each halves its square, while the purple side triangles each quarters its square. The red and purple triangles together therefore comprise $3$ squares, meaning that the shaded region is $\frac{2}{3}$rds of the total square.