The region $C$ is the largest, and we want to minimize $C-(A+B)$,  Since this value can be negative, it means we want $A+B$ to cancel out $C$ and the remaining portion is as large as possible.  Neither $A$ nor $B$ can be larger than $C$, so the best we can do is for one of them equal $C$ to cancel, and maximized the other (still cannot exceed $C$).  This happens when both $A$ and $B$ are equal to $C$, in which case the triangle is equilateral.