球半径R=1
角AOB=π/2,AB=2sin(π/4)=根号2,AC=AB=根号2
角BOC=π/3,BC=2sin(π/6)=1
sin(角BAC/2)=(1/2)/(根号2)=1/(2(根号2))=(根号2)/4
cos(角BAC/2)=(1-((根号2)/4)^2)^(1/2)=(1/4)(根号(14)
sin(角BAC)=2sin(角BAC/2)cos(角BAC/2)=(1/4)(根号7)
设三角形ABC的外接圆半径=r
2r=BC/sin(角BAC)=4/(根号7)
r=2/(根号7)
球心O到平面ABC的距离=(R^2-r^2)^(1/2)=(1-(4/7))^(1/2)=(1/7)(根号21)