Two Maximized Constants in an Inequality

Algebra Level 5

Let $C_1$ be the maximum value such that

$x^2+y^2+1 \geq C_1(x+y)$

for all real $x$ and $y$ . Similarly, let $C_2$ be the maximum value such that

$x^2+y^2+xy+1 \geq C_2(x+y)$

holds true for all real $x$ and $y$ . If $C_1C_2$ can be expressed as $\sqrt{n}$ , find $n$ .

This is really a proof problem in disguise. It's a very beautiful problem, and I hope you post a proof rather than just a brute-force solution explaining your answer.