Random observation I want to make. Let
a = \tan{A}, b = \tan{B}, c = \tan{C}
Then the problem reduces to maximising
\cos(2A) + \cos(2B) + 2\cos C
subject to the constraint
\cos(A+B+C) = 0
and A,B,C are acute angles.
I don't know if this has a nice geometric interpretation, but if anyone...