Very confusing polygon question. Can anyone help? I was practising questions on principles on mathematics. I stumbled onto this question and I don't know where to start. Can anyone please help??
If $P_1P_2....P_n$ is a regular polygon in the $(x,y)$-plane, each side of
length a (so the $P_i$ are the corners of an $n$-sided figure with sides of
equal length $a$). Find the sum
$$
S = (P_1P_2)^2 + (P_1P_3)^2 + \dots + (P_1P_n)^2;
$$
here $P_1P_j$ stands for the length of the line form the point $P_1$ to the
point $P_j$ (your expression for $S$ will be a function of $a$, $n$ and also a well-known trigonometric function).
 A: Let $\zeta\in\mathbb C$ be a primitive $n$th root of unity.
Then in the specaial case that $a=|\zeta-1|$, we have
$$S=\sum_{k=1}^n|\zeta^k-1|^2=\sum_{k=1}^n(2-\zeta^k-\bar\zeta^k)=2n.$$
Hence in general
$$ S = \frac{2a^2n}{|\zeta-1|^2}.$$
Note that $|\zeta-1|=2\sin\frac\pi n$, so that ultimately
$$ S =  \frac{a^2n}{2\sin^2\frac\pi n}.$$
A: The more general situation (and nicer way) to state the problem is: In the circle $\Gamma$ with center $O$ and radius $r$, we have inscribed a regular $n-$gon $P_1 P_2 \ldots P_n$. If $P$ is any point such that $OP=d$, then $\sum | P_i P |^2 = n(r^2+d^2) $.
To show this, you can either use complex numbers, or vectors. I'd present the dot product of vectors.
Let $O$ be the origin. Then $\vec{P_1} + \ldots + \vec{P_n} = 0$. Hence,
$\begin{align}
\sum | P_i P |^2 & = \sum (\vec{P} - \vec{P_i}) \cdot (\vec{P} - \vec{P_i}) \\
& = \sum \left(\vec{P} \cdot \vec{P} + \vec{P} \cdot \vec{P} - 2 \vec{P} \cdot \vec{P_i}\right)\\
& = nd^2 + nr^2 - 2 \vec{P} \cdot \left( \sum \vec{P_i} \right) \\
& = n(d^2+r^2) \\
\end{align}$
Finally, to answer your original question, express $r$ in terms of $a$ and $n$, and set $P=P_1$.
A: 
Consider,
$\theta$= $360/k$
Where $k$ ->Number of sides of polygon.
$a = 2Rsin(\theta$/$2)$ = $2Rsin(360/(2*k))$              
$R = a/(2*sin(180/k))$
$S=(P1P2)^2+(P1P3)^2+⋯+(P1Pn)^2 $= $\sum(2Rsin(180n/k))^2$ 
=$4R^2 \sum sin^2(180*n/k)$                        
= $2R^2(n - cos(n(n-1)180/k))$ = $2 a^2 (n - cos(n(n-1)180/k))/(2*sin(180/k))^2$
$S= a^2 (n - cos(n(n-1)180/k))/(2*sin^2(180/k))$
We have $n = k-1$
$S= a^2 (k-1 - cos((k-1)*(k-2)*180/k)/(2*sin^2(180/k))$
