Mysterious connection

Note that the points $1, e^{2\pi i/17}, e^{4\pi i/17}, \ldots, e^{32\pi i/17}$ form the vertices of a regular heptadecagon inscribed in the unit circle of the complex plane.

If we identify $P_1 = 1$ , then $\begin{aligned}\sum_{n=2}^{17}P_1P_n^2 &= \sum_{n=1}^{16}\left|e^{2n\pi i/17} - 1\right|^2 \\ &= \sum_{n=1}^{16}\left(e^{2n\pi i/17} - 1\right)\left(e^{-2n\pi i/17} - 1\right) \\ &= \sum_{n=1}^{16} \left(2 - e^{2n\pi i/17} - e^{-2n\pi i/17}\right) \\ &= 32 - 2\sum_{n=1}^{16} e^{2n\pi i/17} \\ &= 32 - 2(-1) \\ &= \boxed{34}\end{aligned}$

Call the origin of the unit circle $O$ . Applying the Law of Cosines to triangle $\Delta P_1 O P_n$ we obtain $(P_1P_n)^2=2-2cos(\frac{2(n-1)\pi}{17})$ . So the sum in question is $\sum_{n=2}^{17} (2-2cos(\frac{2(n-1)\pi}{17})) = 32-2\sum_{n=1}^{16} cos(\frac{2n\pi}{17})$ . But we know by symmetry of the cosine function that $\sum_{n=1}^{17} cos(\frac{2n\pi}{17})=0$ (i.e. using the unit circle), and so $\sum_{n=1}^{16} cos(\frac{2n\pi}{17})=-1$ . Thus the sum in question evaluates to $\boxed{34}$ .