Let $A_1, A_2, A_3, ……. , A_n$ be vertices of a regular polygon of $n$ sides circumscribed about a circle whose centre is O and radius is $a$. P is any point other than O inside the circle. Then show that sum of squares from P to the sides of polygon is $n(a^2 + \dfrac{c^2}{2})$, where OP = c.
I just need to prove that $(a + c)^2 cos^2{\dfrac{\pi}{n}} = a^2 + \dfrac{c^2}{2}$. How shall I prove that
Best Answer
HINT.
If $S$ is the sum of the squared distances from $P$ to the sides of the polygon, then: $$ S=\sum_{k=0}^{n-1}\left(a-c\cdot\cos\left(\phi+{2\pi\over n}k\right)\right)^2= na^2+c^2\sum_{k=0}^{n-1}\cos^2\left(\phi+{2\pi\over n}k\right) -2ac\sum_{k=0}^{n-1}\cos\left(\phi+{2\pi\over n}k\right). $$ But: $$\cos^2\alpha={1+\cos2\alpha\over2} \quad\text{and}\quad \sum_{k=0}^{n-1}\cos\left(\phi+{2\pi\over n}k\right)=0. $$