Question Number 173078 by AgniMath last updated on 06/Jul/22
Answered by som(math1967) last updated on 07/Jul/22
$$\boldsymbol{{let}}\:\boldsymbol{{b}}−\boldsymbol{{c}}=\boldsymbol{{x}}\:,\left(\boldsymbol{{a}}−\boldsymbol{{b}}\right)=\boldsymbol{{y}},\boldsymbol{{c}}−\boldsymbol{{a}}=\boldsymbol{{z}} \\ $$$$\therefore\:\boldsymbol{{x}}+\boldsymbol{{y}}+\boldsymbol{{z}}=\mathrm{0} \\ $$$$\left(\boldsymbol{{x}}+\boldsymbol{{y}}\right)^{\mathrm{2}} =\left(−\boldsymbol{{z}}\right)^{\mathrm{2}} \\ $$$$\:\boldsymbol{{x}}^{\mathrm{2}} +\boldsymbol{{y}}^{\mathrm{2}} −\boldsymbol{{z}}^{\mathrm{2}} =−\mathrm{2}\boldsymbol{{xy}} \\ $$$$\Rightarrow\left(\boldsymbol{{x}}^{\mathrm{2}} +\boldsymbol{{y}}^{\mathrm{2}} −\boldsymbol{{z}}^{\mathrm{2}} \right)^{\mathrm{2}} =\mathrm{4}\boldsymbol{{x}}^{\mathrm{2}} \boldsymbol{{y}}^{\mathrm{2}} \\ $$$$\Rightarrow\boldsymbol{{x}}^{\mathrm{4}} +\boldsymbol{{y}}^{\mathrm{4}} +\boldsymbol{{z}}^{\mathrm{4}} =\mathrm{4}\boldsymbol{{x}}^{\mathrm{2}} \boldsymbol{{y}}^{\mathrm{2}} −\mathrm{2}\boldsymbol{{x}}^{\mathrm{2}} \boldsymbol{{y}}^{\mathrm{2}} +\mathrm{2}\boldsymbol{{y}}^{\mathrm{2}} \boldsymbol{{z}}^{\mathrm{2}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:+\mathrm{2}\boldsymbol{{z}}^{\mathrm{2}} \boldsymbol{{x}}^{\mathrm{2}} \\ $$$$\Rightarrow{x}^{\mathrm{4}} +{y}^{\mathrm{4}} +\boldsymbol{{z}}^{\mathrm{4}} =\mathrm{2}\left(\boldsymbol{{x}}^{\mathrm{2}} \boldsymbol{{y}}^{\mathrm{2}} +\boldsymbol{{y}}^{\mathrm{2}} \boldsymbol{{z}}^{\mathrm{2}} +\boldsymbol{{z}}^{\mathrm{2}} \boldsymbol{{x}}^{\mathrm{2}} \right) \\ $$$$\Rightarrow\frac{\mathrm{1}}{\mathrm{2}}\left(\boldsymbol{{x}}^{\mathrm{4}} +\boldsymbol{{y}}^{\mathrm{4}} +\boldsymbol{{z}}^{\mathrm{4}} \right)=\left(\boldsymbol{{xy}}\right)^{\mathrm{2}} +\left(\boldsymbol{{yz}}\right)^{\mathrm{2}} +\left(\boldsymbol{{zx}}\right)^{\mathrm{2}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:+\mathrm{2}\boldsymbol{{xyz}}\left(\boldsymbol{{x}}+\boldsymbol{{y}}+\boldsymbol{{z}}\right) \\ $$$$\:\:\:\left[\because\:\boldsymbol{{x}}+\boldsymbol{{y}}+\boldsymbol{{z}}=\mathrm{0}\right] \\ $$$$\frac{\mathrm{1}}{\mathrm{2}}\left(\boldsymbol{{x}}^{\mathrm{4}} +\boldsymbol{{y}}^{\mathrm{4}} +\boldsymbol{{z}}^{\mathrm{4}} \right)=\left(\boldsymbol{{xy}}\right)^{\mathrm{2}} +\left(\boldsymbol{{yz}}\right)^{\mathrm{2}} +\left(\boldsymbol{{zx}}\right)^{\mathrm{2}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{2}\boldsymbol{{xy}}.\boldsymbol{{yz}}+\mathrm{2}\boldsymbol{{xzyz}}+\mathrm{2}\boldsymbol{{xy}}.\boldsymbol{{zx}} \\ $$$$\Rightarrow\frac{\mathrm{1}}{\mathrm{2}}\left(\boldsymbol{{x}}^{\mathrm{4}} +\boldsymbol{{y}}^{\mathrm{4}} +\boldsymbol{{z}}^{\mathrm{4}} \right)=\left(\boldsymbol{{xy}}+\boldsymbol{{yz}}+\boldsymbol{{zx}}\right)^{\mathrm{2}} \\ $$$$\therefore\frac{\mathrm{1}}{\mathrm{2}}\left\{\left(\boldsymbol{{b}}−\boldsymbol{{c}}\right)^{\mathrm{4}} +\left(\boldsymbol{{c}}−\boldsymbol{{a}}\right)^{\mathrm{4}} +\left(\boldsymbol{{a}}−\boldsymbol{{b}}\right)^{\mathrm{4}} \right\}= \\ $$$$\:\:\left\{\left(\boldsymbol{{b}}−\boldsymbol{{c}}\right)\left(\boldsymbol{{a}}−\boldsymbol{{b}}\right)+\left(\boldsymbol{{c}}−\boldsymbol{{a}}\right)\left(\boldsymbol{{a}}−\boldsymbol{{b}}\right)+\left(\boldsymbol{{b}}−\boldsymbol{{c}}\right)\left(\boldsymbol{{c}}−\boldsymbol{{a}}\right)\right\}^{\mathrm{2}} \\ $$$$\: \\ $$$$ \\ $$
Commented by Tawa11 last updated on 11/Jul/22
$$\mathrm{Great}\:\mathrm{sir} \\ $$