# 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?

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