I think we can as well solve this way...

assume G be the centroid..

Let <GBC=B1 and <GBA=B2  and GB=x GC=y

 

now from rt triangles BDG and BGC, we can find cot<B1 and cot<B2

apply compound angle formula for cot<B1+B2

similarly for<C and adding both we get an algebraic eqn in x,y

hence we can find its minimum....

this has no restrictions for <B and <C

                                                                         goutham