# Think you can Geometry?

**Geometry**Level pending

A regular polygon of with an even number of sides is inscribed in a circle of radius \(5\). The sum of the squares of the distances from vertex \(A\) to all the other vertices of the polygon is \(800\). How many sides does the polygon have?. Hint: Polar coordinates and roots of unity! WLOG, let the point A we want to use be \((5, 0)\). Let \(5\text{cis}\theta, 5\text{cis} 2\theta, . . . 5\text{cis}(2n-1)\theta\) be the other vertices of the polygon. What does the distance from \(A\) to \(5\text{cis} \theta\) squared plus the distance from \(A\) to \(5\text{cis} (n + 1)\theta\) squared equal?