Irrationality of π

Lambert's theorem. For $x\in\mathbb{Q}\setminus\{0\}$, $\tan x$ is irrational.

Application. $\tan(\pi/4)=1\in\mathbb{Q}$. If $\pi/4\in\mathbb{Q}$, this would contradict Lambert (since $\pi/4\neq0$). So $\pi/4\notin\mathbb{Q}$.

Conclusion. $\pi$ is irrational.

Remark. The first proof was due to Lambert (1761). Lindemann later proved $\pi$ is transcendental (1882).