Lesser known derivations of well-known formulas and theorems

What are some lesser known derivations of well-known formulas and theorems?

I ask because I recently found a new way to derive the quadratic formula which didn't involve completing the square as is commonly taught. Doing so I was wondering what other proofs and derivations for other formulas that have remained unknown to most people? Whether it be because the proof is too complex or less pretty, I still find it insightful to see different ways to solve a problem. To me it makes me understand proofs better, and thus also giving a better comprehension of it.

$$ \begin{align} &\text{Given a quadratic function } f:\\[0.1em] f &=ax^2+bx+c = a(x-r_1)(x-r_2) = ax^2-a(r_1+r_2)+ar_1r_2\\[0.1em] a &= a,\enspace \frac{b}{a} = -(r_1+r_2),\enspace \frac{c}{a} = r_1r_2\\[1em] f' &= 2ax+b, \enspace f'(x) = 0 \Rightarrow x = -\frac{b}{2a}\\[0.2em] \text{This is an}& \text{ extremum of } f \text{, and is equidistant from each root } r_1, \enspace r_2 \text{ as shown:}\\[0.4em] \frac{b}{a} &= -(r_1+r_2) \iff -\frac{b}{2a} = \frac{r_1+r_2}{2} \\[1em] \Rightarrow \enspace &\text{The roots are of the form } r= -\frac{b}{2a}\pm d\\[1em] \frac{c}{a} = r_1r_2 &= (-\frac{b}{2a}+d)(-\frac{b}{2a}-d) = \frac{b^2}{4a^2}-d^2\\[0.2em] \Rightarrow\enspace& d^2 = \frac{b^2}{4a^2}-\frac{c}{a} = \frac{b^2}{4a^2}-\frac{4ac}{4a^2} = \frac{b^2-4ac}{4a^2}\\[0.2em] \Rightarrow\enspace&d = \pm\frac{\sqrt{b^2-4ac}}{2a}\\[1em] \text{Which yields }& r = -\frac{b}{2a}\pm d = \frac{-b\pm\sqrt{b^2-4ac}}{2a}\enspace\square \end{align} $$

The part of Wilson's theorem which states that $(p-1)!\equiv -1 {\mod p}$ for any prime $p$ is normally proved by grouping elements in the product $(p-1)!$ with their inverses, but it also admits a proof using Sylow theory by first showing that there are $(p-2)!$ Sylow $p$-subgroups of $S_p$. To do this, observe first that a Sylow $p$-subgroup of $S_p$ is generated by a permutation of $p$ objects, of order $ p $. There are $(p-1)!$ of these, and each subgroup is isomorphic to a cyclic group with $p$ elements, which has $p-1$ generators.

By the third Sylow theorem, we then have $(p-2)!\equiv 1 {\mod p}$ and the result follows.

In 1955, Hillel Furstenberg gave a topological proof for the fact that there are infinitely many prime numbers which derives a contradiction by observing that if there were only finitely many, a certain set would be closed that from topological arguments is known not to be closed.

The full proof can be found here.

Here's a proof due to Zagier that every prime $p = 4n + 1$ is the sum of two squares: On the finite set $A = \{(x, y, z)\in \mathbb{Z}^{\geq 0}:\, p = x^2 + 4yz\}$, define $$f(x, y, z) = \begin{cases} (x + 2z, z, y - x - z) & \text{if $x < y - z$}; \\ (2y - x, y, x - y + z) & \text{if $y - z < x < 2y$}; \\ (x - 2y, x -y + z, y) & \text{if $x > 2y$}. \end{cases}$$ The function $f$ has exactly one fixed point: $(1, 1, n)$. Any involution $g$ of $A$ (i.e., function with $g = g^{-1}$) must have a number of fixed points $F_g$ equal to $\#A \pmod{2}$, since $\frac{1}{2}\left(F_g + \#A\right)$ is an integer by Burnside's theorem. (Specifically, it's the number of orbits of $A$ under $g$). Thus the involution $(x, y, z) \to (x, z, y)$ must have also have an odd number of fixed points; in particular, there must exist at least one fixed point $(a, b, b)\in A$. Thus $p = a^2 + (2b)^2$, as required.

The Cantor-Bernstein theorem is most commonly proved using an argument that involves the integers. In fact, there is a simple proof based on an elementary fixed point lemma.