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Question Number 145246 by gsk2684 last updated on 03/Jul/21

$$\mathrm{prove}\mathrm{that}\mathrm{a}\mathrm{triangle}\mathrm{inscribed}$$$$\mathrm{in}\mathrm{a}\mathrm{circle}\mathrm{of}\mathrm{radius}\mathrm{r}\mathrm{having}\mathrm{maximum}$$$$\mathrm{area}\mathrm{is}\mathrm{an}\mathrm{equilateral}\mathrm{triangle}\mathrm{with}$$$$\mathrm{side}\sqrt{3}\mathrm{r}.$$

Answered by Olaf_Thorendsen last updated on 03/Jul/21

$$\mathrm{Necessarily},\mathrm{by}\mathrm{symmetry}a=b=c$$$${\mathrm{AB}}^2={\mathrm{OA}}^2+{\mathrm{OB}}^2-2\mathrm{OA}.\mathrm{OB}.\cos 120°$$$$a^2=r^2+r^2-2r.r(-\frac{1}{2})$$$$a^2=3r^2\Rightarrow a=\sqrt{3}r$$$$\mathrm{The}\mathrm{triangle}\mathrm{is}\mathrm{equilateral}\mathrm{and}:$$$$\mathrm{S}=\frac{\sqrt{3}}{4}{a}^2=\frac{\sqrt{3}}{4}\left(3r^2\right)=\frac{3\sqrt{3}}{4}{r}^2$$

Commented by gsk2684 last updated on 03/Jul/21

$$\mathrm{thank}\mathrm{you}.$$$$\mathrm{Will}\mathrm{you}\mathrm{please}\mathrm{explain}\mathrm{how}\mathrm{to}\mathrm{prove}$$$$\mathrm{to}\mathrm{prove}\mathrm{required}\mathrm{triangle}\mathrm{is}$$$$\mathrm{an}\mathrm{equilateral}\mathrm{triangle}$$

Commented by ajfour last updated on 03/Jul/21

$$circleiscircular.$$

Commented by peter frank last updated on 04/Jul/21

$$thankyou.canyougivea$$$$diagram$$

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