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Question Number 94296 by mathocean1 last updated on 17/May/20

Exercise  ABC is a triangle. A′ ; B′ ; C′ are respec−  tively middles of sides: [BC]; [AC] and [AB].  G is isobarycenter( situated at equal distance  ) of A, G , and C.  1) By using the theorem of medians,  show that:  GB^2 +GC^2 =(1/2)GA^2 +(1/2)BC^2

$$\mathrm{Exercise} \\ $$$$\mathrm{ABC}\:\mathrm{is}\:\mathrm{a}\:\mathrm{triangle}.\:\mathrm{A}'\:;\:\mathrm{B}'\:;\:\mathrm{C}'\:\mathrm{are}\:\mathrm{respec}− \\ $$$$\mathrm{tively}\:\mathrm{middles}\:\mathrm{of}\:\mathrm{sides}:\:\left[\mathrm{BC}\right];\:\left[\mathrm{AC}\right]\:\mathrm{and}\:\left[\mathrm{AB}\right]. \\ $$$$\mathrm{G}\:\mathrm{is}\:\mathrm{isobarycenter}\left(\:\mathrm{situated}\:\mathrm{at}\:\mathrm{equal}\:\mathrm{distance}\right. \\ $$$$\left.\right)\:\mathrm{of}\:\mathrm{A},\:\mathrm{G}\:,\:\mathrm{and}\:\mathrm{C}. \\ $$$$\left.\mathrm{1}\right)\:\mathrm{By}\:\mathrm{using}\:\mathrm{the}\:\mathrm{theorem}\:\mathrm{of}\:\mathrm{medians}, \\ $$$$\mathrm{show}\:\mathrm{that}: \\ $$$$\mathrm{GB}^{\mathrm{2}} +\mathrm{GC}^{\mathrm{2}} =\frac{\mathrm{1}}{\mathrm{2}}\mathrm{GA}^{\mathrm{2}} +\frac{\mathrm{1}}{\mathrm{2}}\mathrm{BC}^{\mathrm{2}} \\ $$

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