Prove that in an acute triangle the circumcenter falls in the interior of the triangle?

Question

How can we prove the above theorem using synthetic geometry.

Answer 1

Suppose otherwise, that the circumcenter $O$ of $\Delta ABC$ lies outside the triangle. Suppose that $BC$ is the closest side to $O$. Let $CD$ be a diameter. We have $\angle BAC>\angle DAC=90^\circ$, contradicting the fact that $\Delta ABC$ is acute. Hence, $O$ must lie inside $\Delta ABC$.