过点A作BC垂线,垂足为E
根据勾股定理有：
AB^2=BE^2+AE^2
AC^2=AE^2+EC^2
AE^2=AO^2-OE^2
O是BC中点,所以BO=OC
BE=BO+OE=OC+OE
EC=OC-OE
AB^2+AC^2=BE^2+AE^2+AE^2+EC^2
=2AE^2+BE^2+EC^2
=2(AO^2-OE^2)+(OC+OE)^2+(OC-OE)^2
=2*AO^2-2*OE^2+OC^2+2*OC*OE+OE^2+OC^2-2*OC*OE+OE^2
=2*AO^2+2*OC^2
=2(AO^2+OC^2)