# Solution to the [[Three Squares V]] Puzzle +-- {.image} [[ThreeSquaresV.jpeg:pic]] > Three squares. What’s the angle? =-- ## Solution by [[Angle at the Circumference is Half the Angle at the Centre]] +-- {.image} [[ThreeSquaresVAnnotated.jpeg:pic]] =-- 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]] +-- {.image} [[ThreeSquaresVInvariance.jpeg:pic]] =-- 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 $45^\circ$.