Hermite Constant for Dimension 2

Theorem
The Hermite constant for dimension $2$ is:


 * $\gamma_2 = \dfrac 2 {\sqrt 3}$

or, as it is often presented:


 * $\paren {\gamma_2}^2 = \dfrac 4 3$

Proof
The statement of the result to be proved can be expressed as:


 * There exist non-zero $x$ and $y$ such that:
 * $\paren {a x^2 + 2 b x y + c y^2}^2 \le \size {a c - b^2} \times \dfrac 4 3$