Notes
two quarter circles in a rectangle solution

Solution to the Two Quarter Circles in a Rectangle Puzzle

Solution by Isosceles Triangles, Angles in a Triangle, and Properties of Circles

With the points labelled as above, the point where the quarter circles touch, $F$, lies on the line between their centres, so $B F A$ is a straight line and is a diagonal of the rectangle.

As $A$ is the centre of the red quarter circle, triangle $A F E$ is isosceles and therefore angle $A \hat{F} E$ is the same as angle $A \hat{E} F$. Similarly, angle $B \hat{F} C$ is the same as angle $B \hat{C} F$.

Since angles in a triangle add up to $180^\circ$, the angles in triangle $A F E$ and in triangle $B C F$ add up together to $360^\circ$. So

Then also $A B D$ is a triangle and angle $A \hat{D} B$ is a right-angle, so $E \hat{A} F + C \hat{B} F = 90^\circ$, meaning that: