Notes
zig-zag inside a rectangle solution

Solution to the Zig-zag Inside a Rectangle Puzzle

Solution by Similar Triangles, Angles in a Triangle, Angles in a Semi-Circle, and Angles in the Same Segment

In the above diagram, the orange circle is drawn so that $H D$ is a diameter.

The purple rectangles are similar, so triangles $H I B$ and $B I D$ are also similar, with the vertices corresponding in that order. So angles $H \hat{B} I$ and $B \hat{D} I$ are the same. Since angle $B \hat{I} D$ is a right-angle, this means that angles $I \hat{B} D$ and $H \hat{B} I$ add up to $90^\circ$, so angle $H \hat{B} D$ is a right-angle. Since the angle in a semi-circle is a right-angle, this means that $B$ lies on the orange circle.

By a similar argument, so also $F$ lies on the orange circle.

As the blue rectangles are similar, angles $F \hat{D} E$ and $F \hat{H} J$ are the same. Since angles in the same segment are equal, the angle $F \hat{B} D$ is also the same as $F \hat{H} J$ and as $F \hat{D} E$. Lastly, triangles $A B H$ and $I H B$ are congruent so angles $A \hat{H} B$ and $H \hat{B} I$ are equal, so the three marked angles add up to angle $H \hat{B} D$ which is $90^\circ$.