Two squares and three line segments. What’s the angle α?

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## Solution

The angle α is 90°.

We define the vectors **a** and **b** along the respective square sides, as well as R**a** and R**b**, where R is an anticlockwise rotation over 90°. Now consider the vector to the midpoint: it is (**a**+R**b**)/2. The vector connecting the lower vertices is **b**-R**a**.

Taking their dot product we get (**a**+R**b**)**·**(** b**-R

**)/2=(**

**a****a·b**-R

**b·**R

**a**–

**a·**R

**a**+R

**b·b**)/2. Now the last two terms are 0, because these vectors are perpendicular. We also know that a rotation does not alter the (commutative) dot product, so the first two terms cancel each other. The total dot product is therefore 0, implying that α is right angle.