Notes
three squares v solution

Solution to the Three Squares V Puzzle

Solution by Angle at the Circumference is Half the Angle at the Centre

With the points labelled as above, the length of $O C$ is the same as $O A$, hence also the same as $O B$. Therefore a circle with centre $O$ that passes through $A$ also passes through $B$ and $C$. Then since the angle at the circumference is half the angle at the centre, angle $B \hat{C} A$ is half of angle $B \hat{O} A$, which is $90^\circ$ as it is the corner of a square. Hence angle $B \hat{C} A$ is $45^\circ$.

Solution by Invariance Principle

The purple squares can be tilted, providing $O$ remains on the joining line. With the configuration above, the purple squares are the same size as the blue square, and so the triangle $O B C$ is an isosceles right-angled triangle so angle $B \hat{C} A$ is $45^\circ$.