Notes
three triangles and a rectangle inside a circle solution

Solution to the Three Triangles and a Rectangle Inside a Circle Puzzle

Solution by Symmetry and Angle in a Semi-Circle

In the diagram above, the vertical line through the midpoint of chord $B C$ passes through the centre of the circle. It is therefore a line of symmetry of both the circle and the central equilateral triangle. Since the points $A$ and $D$ depend on the circle and the central equilateral triangle, reflecting in the vertical line swaps $A$ and $D$ and so in particular the line joining $A$ to $D$ is the continuation of their horizontal sides. This establishes angle $D \hat{A} E$ as a right-angle, and so $E A$ is a diameter of the circle since the angle in a semi-circle is a right-angle. By the same result, angle $E \hat{D} F$ is then also a right-angle.