>>786
MN=^OM+ON=-OA/2+(OB+OC)/2
OG*MN={(OA+OB+OC)/3}*{-OA/2+(OB+OC)/2}
=(1/6)(OA+OB+OC)*(-OA+OB+OC)
=(1/6)(-lOAl^2+(OB+OC)^2)
=(1/6)(-4+lOBl^2+2OB*OC+lOCl^2)
=(1/6)(-4+4+2*2*2cosπ/3+4)
=4/3

①
AB=DC=EF=HG=a
AD=BC=EH=FG=b
AE=BF=CG=DH=c
とする。
a*b=b*c=0,c*a=1/2
AI=AB+BI=AB+tBD=a+tc
lAIl^2=(a+tc)^2=lal^2+2ta*c+t^2lcl^2
=1+2t*(1/2)+t^2*1
=t^2+t+1
lAIl=√(t^2+t+1)

②
AG=AB+BC+CG=a+b+c
lAGl^2=(a+b+c)^2=1+1+1+2a*b+2b*c+2c*a=1+1+1+2*1*1cosπ/3=4
AG*AI=(a+b+c)*(a+tc)=1+1/2+t/2+t=(3/2)(t+1)
△AGI=(1/2)AG*AIsin∠IAG
2(△AGI)^2=(AG*AIsin∠IAG)^2=lAGl^2lAIl^2-(AG*AI)^2
=4(t^2+t+1)-{(3/2)(t+1)}^2
=・・・
=(7/16){(t-1/7)^2+48/49}
△AGIの最小値はt=1/7の時