# Solution to the Three Overlapping Squares Puzzle +-- {.image} [[ThreeOverlappingSquares.png:pic]] > The side lengths of the two blue squares are given. What's the missing length? =-- ## Solution by [[Angle at the Centre is Twice the Angle at the Circumference]] +-- {.image} [[ThreeOverlappingSquaresLabelled.png:pic]] =-- Since the diagonal of the dark blue rectangle lies along the top edge of the black square, angle $A \hat{B} C$ is $135^\circ$. The reflex angle $A \hat{O} C$ is $360^\circ - 90^\circ = 270^\circ$. Since $270^\circ$ is twice $135^\circ$, using the fact that the [[angle at the centre is twice the angle at the circumference]], the circle centred at $O$ through $A$ and $C$ also passes through $B$. Then $O B$ has the same length as $O A$ and so has length $7$. ## Solution by [[Invariance Principle]] +-- {.image} [[ThreeOverlappingSquaresSpecial.png:pic]] =-- In this configuration, the points $A$ and $B$ coincide and so the requested length is a side of the square, which is $7$.