Dirac's Theorem

Theorem
If a connected graph $G$ has $n \ge 3$ vertices and the degree of each vertex is at least $\dfrac n 2$, then $G$ is Hamiltonian.

Proof
Take any two non-adjacent vertices $u, v \in G$.

Then:
 * $\displaystyle \deg u + \deg v \ge \frac n 2 + \frac n 2 = n$

The result follows by a direct application of Ore's Theorem.