Question Number 201526 by 281981 last updated on 08/Dec/23
Answered by AST last updated on 08/Dec/23
$${WLOG},{let}\:{O}\:{be}\:{the}\:{origin};\:{g}=\frac{{a}+{b}+{c}}{\mathrm{3}} \\ $$$$\mid{o}−{a}\mid=\mid{o}−{b}\mid=\mid{o}−{c}\mid={R} \\ $$$$\Rightarrow\left({o}−{a}\right)\left(\overset{−} {{o}}−\overset{−} {{a}}\right)={R}^{\mathrm{2}} \Rightarrow{a}\overset{−} {{a}}={R}^{\mathrm{2}} \Rightarrow\overset{−} {{a}}=\frac{{R}^{\mathrm{2}} }{{a}} \\ $$$${Then}\:\mid{OG}\mid^{\mathrm{2}} =\left({g}−{o}\right)\left(\overset{−} {{g}}−\overset{−} {{o}}\right)={g}\overset{−} {{g}} \\ $$$$=\left(\frac{{a}+{b}+{c}}{\mathrm{3}}\right)\left(\frac{{R}^{\mathrm{2}} \left(\frac{\mathrm{1}}{{a}}+\frac{\mathrm{1}}{{b}}+\frac{\mathrm{1}}{{c}}\right)}{\mathrm{3}}\right)=\frac{{R}^{\mathrm{2}} \left({a}+{b}+{c}\right)\left({ab}+{bc}+{ca}\right)}{\mathrm{9}{abc}} \\ $$$$\mid{b}−{a}\mid^{\mathrm{2}} +\mid{b}−{c}\mid^{\mathrm{2}} +\mid{c}−{a}\mid^{\mathrm{2}} \\ $$$$={R}^{\mathrm{2}} \left({b}−{a}\right)\left(\frac{{a}−{b}}{{ab}}\right)+{R}^{\mathrm{2}} \left({b}−{c}\right)\left(\frac{{c}−{b}}{{bc}}\right)+{R}^{\mathrm{2}} \left({c}−{a}\right)\left(\frac{{a}−{c}}{{ac}}\right) \\ $$$$=\frac{{R}^{\mathrm{2}} }{{abc}}\left[{c}\left({b}−{a}\right)\left({a}−{b}\right)+{a}\left({b}−{c}\right)\left({c}−{b}\right)+{b}\left({c}−{a}\right)\left({a}−{c}\right)\right] \\ $$$$=\frac{{r}^{\mathrm{2}} }{{abc}}\left[\left\{−\mathrm{3}{abc}+\mathrm{3}{abc}−{a}^{\mathrm{2}} {c}−{b}^{\mathrm{2}} {c}−{ab}^{\mathrm{2}} −{c}^{\mathrm{2}} {a}+\left({abc}−{a}^{\mathrm{2}} {b}\right)−{c}^{\mathrm{2}} {b}+\mathrm{6}{abc}\right]\right. \\ $$$$=\frac{−{R}^{\mathrm{2}} }{{abc}}\left[{a}^{\mathrm{2}} {b}+{abc}+{a}^{\mathrm{2}} {c}+{ab}^{\mathrm{2}} +{b}^{\mathrm{2}} {c}+{abc}+{abc}+{bc}^{\mathrm{2}} +{c}^{\mathrm{2}} {a}−\mathrm{9}{abc}\right] \\ $$$$=\frac{−{r}^{\mathrm{2}} \left({a}+{b}+{c}\right)\left({ab}+{bc}+{ca}\right)}{{abc}}+\mathrm{9}{r}^{\mathrm{2}} \\ $$$$\Rightarrow\mid{BA}\mid^{\mathrm{2}} +\mid{AC}\mid^{\mathrm{2}} +\mid{CB}\mid^{\mathrm{2}} =−\mathrm{9}\mid{OG}\mid^{\mathrm{2}} +\mathrm{9}{R}^{\mathrm{2}} \\ $$$$\left.\Rightarrow\mathrm{3}\right) \\ $$