Sum of Squares from a Symmetric System

If $(a+b)(c+b) = 2$, $(a+c)(b+c) = 3$, and $(b+a)(c+a) = 4$, what is the value of $a^2 + b^2 + c^2$? The answer is in the form $\frac{\alpha}{\beta}$ where $\alpha$ and $\beta$ are coprime positive integers. Input the value of $\alpha + \beta$.