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Question Number 175756 by mnjuly1970 last updated on 06/Sep/22

Commented by mr W last updated on 06/Sep/22

$$ithinkforanytypeoftriangle,$$$$\frac{r}{R}ismaximumwhenA=B=C=60°.$$

Answered by mr W last updated on 06/Sep/22

$$\alpha =\mathrm{\angle }A$$$$a=BC$$$$R=\frac{a}{2\sin \alpha }$$$$r=\frac{a}{2}\times \tan \left(\frac{\frac{\pi }{2}-\frac{\alpha }{2}}{2}\right)=\frac{a}{2}\times \tan (\frac{\pi }{4}-\frac{\alpha }{4})$$$$\frac{r}{R}=\lambda =\tan (\frac{\pi }{4}-\frac{\alpha }{4})\sin \alpha$$$$\frac{d\lambda }{d\alpha }=0\Rightarrow \alpha =\frac{\pi }{3}$$

Answered by behi834171 last updated on 06/Sep/22

$$maxof:\frac{R}{\mathit{r}}=2\Rightarrow accordingto{euler}^{\prime }srule$$$${\mathit{d}}^2={\mathit{R}}^2-2\mathit{R}.\mathit{r}\stackrel{\mathit{R}=2\mathit{r}}{\Rightarrow }\mathit{d}=0$$$$\mathit{d},is\mathit{d}\mathit{i}\mathit{s}\mathit{t}\mathit{a}\mathit{n}\mathit{c}\mathit{e}\mathit{a}\mathit{m}\mathit{o}\mathit{n}\mathit{g}\mathit{c}\mathit{e}\mathit{n}\mathit{t}\mathit{e}\mathit{r}\mathit{s}\mathit{o}\mathit{f}$$$$\mathit{i}\mathit{n}\mathit{c}\mathit{i}\mathit{r}\mathit{c}\mathit{l}\mathit{e}\mathit{a}\mathit{n}\mathit{d}\mathit{o}\mathit{u}\mathit{t}\mathit{c}\mathit{i}\mathit{r}\mathit{c}\mathit{l}\mathit{e}.$$$$\mathit{o}\mathit{n}\mathit{l}\mathit{y}\mathit{i}\mathit{n}\mathit{e}\mathit{q}\mathit{u}\mathit{i}\mathit{l}\mathit{a}\mathit{t}\mathit{e}\mathit{r}\mathit{a}\mathit{l}\mathit{t}\mathit{r}\mathit{i}\mathit{a}\mathit{n}\mathit{g}\mathit{l}\mathit{e}\mathit{d}=0$$$$\mathit{s}\mathit{o}:\mathrm{\angle }A=\mathrm{\angle }B=\mathrm{\angle }C={60}^{\bullet }.◼$$

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