A problem in Steele's Cauchy Schwarz Master Class asks the reader to prove these "variants of the polarization identity".
Let $\langle \cdot, \cdot \rangle$ be a complex inner product and $\alpha \in \mathbb{C}$ with $\alpha^N = 1$ but $\alpha^2 \neq 1$. Then $$ \langle x,y \rangle = \frac{1}{N} \sum_{n=0}^{N-1} \left\|x+\alpha^n y \right\|^2 \alpha^n \quad \text{and} \quad \langle x,y \rangle = \frac{1}{2\pi} \int_{-\pi}^\pi \left\|x+e^{i\theta}y \right\|^2 e^{i \theta} \ d\theta.$$
I can prove it, so I'll pose the more important question(s): How does one find these? Is there any intuitive (e.g. geometric) interpretation? Where in mathematics do these come up?