Question Number 206093 by mr W last updated on 06/Apr/24
Answered by A5T last updated on 07/Apr/24
Commented by A5T last updated on 07/Apr/24
$${S}_{\mathrm{4}} +{S}_{\mathrm{1}} +{S}_{\mathrm{5}} +{S}_{\mathrm{2}} +{S}_{\mathrm{6}} +{S}_{\mathrm{3}} ={S}+{S}_{\mathrm{7}} +{S}_{\mathrm{8}} …\left({i}\right) \\ $$$${S}_{\mathrm{4}} +{S}_{\mathrm{7}} +{S}_{\mathrm{3}} +{S}_{\mathrm{1}} +{S}_{\mathrm{8}} +{S}_{\mathrm{2}} ={S}+{S}_{\mathrm{5}} +{S}_{\mathrm{6}} …\left({ii}\right) \\ $$$$\left({i}\right)−\left({ii}\right):\:{S}_{\mathrm{5}} +{S}_{\mathrm{6}} −{S}_{\mathrm{7}} −{S}_{\mathrm{8}} ={S}_{\mathrm{7}} +{S}_{\mathrm{8}} −{S}_{\mathrm{5}} −{S}_{\mathrm{6}} \\ $$$$\Rightarrow{S}_{\mathrm{5}} +{S}_{\mathrm{6}} ={S}_{\mathrm{7}} +{S}_{\mathrm{8}} \:{into}\:\left({i}\right) \\ $$$$\Rightarrow{S}_{\mathrm{4}} +{S}_{\mathrm{1}} +{S}_{\mathrm{2}} +{S}_{\mathrm{3}} ={S} \\ $$
Commented by mr W last updated on 07/Apr/24
$${yes}\:{sir}!\:{thanks}! \\ $$
Answered by mr W last updated on 07/Apr/24
Commented by mr W last updated on 07/Apr/24
$$\Delta{BCG}=\Delta{BGD} \\ $$$$\Delta{DAE}=\Delta{DEB} \\ $$$$\Delta{BCG}+\Delta{DAE}=\Delta{BGD}+\Delta{DEB} \\ $$$$\Rightarrow{S}_{\mathrm{1}} +{S}_{\mathrm{5}} +{S}_{\mathrm{2}} +{S}_{\mathrm{3}} +{S}_{\mathrm{7}} +{S}_{\mathrm{4}} ={S}_{\mathrm{6}} +{S}+{S}_{\mathrm{8}} \:\:\:…\left({i}\right) \\ $$$${similarly} \\ $$$${S}_{\mathrm{1}} +{S}_{\mathrm{8}} +{S}_{\mathrm{4}} +{S}_{\mathrm{2}} +{S}_{\mathrm{6}} +{S}_{\mathrm{3}} ={S}_{\mathrm{5}} +{S}+{S}_{\mathrm{7}} \:\:\:…\left({ii}\right) \\ $$$$\left({i}\right)+\left({ii}\right): \\ $$$$\mathrm{2}\left({S}_{\mathrm{1}} +{S}_{\mathrm{2}} +{S}_{\mathrm{3}} +{S}_{\mathrm{4}} \right)=\mathrm{2}{S} \\ $$$$\Rightarrow{S}={S}_{\mathrm{1}} +{S}_{\mathrm{2}} +{S}_{\mathrm{3}} +{S}_{\mathrm{4}} \:\checkmark \\ $$