|AB|=c、|AC|=b、|AO|=R
AB*AC=bccosA=bc/3
AO*AB=αAB^2+βAB*AC即R*c*[(c/2)/R]=αc^2+βbc/3即3c=6αc+2βb.①
AO*AC=αAB*AC+βAC^2即R*b*[(c/2)/R]=αbc/3+βb^2即3b=2αc+6βb.②
α=(9c-3b)/16c=9/16-3b/16c
β=(9b-3c)/16b=9/16-3c/16b
α+β=9/8-3(b/c+c/b)/16≤9/8-3*2/16=3/4
(当且仅当b=c即|AB|=|AC|时等号成立)