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Question Number 18847 by Tinkutara last updated on 30/Jul/17

$$\mathrm{In}\mathrm{an}\mathrm{equilateral}\mathrm{triangle}\mathrm{with}\mathrm{usual}$$$$\mathrm{notations}\mathrm{the}\mathrm{value}\mathrm{of}\frac{27r^{2}R}{r_1{r}_2{r}_3}\mathrm{is}\mathrm{equal}\mathrm{to}$$

Answered by behi.8.3.4.1.7@gmail.com last updated on 31/Jul/17

$$inequilateraltriangle:$$$$1)R=2r=\frac{a}{2sinA}=\frac{a}{2\times \frac{\sqrt{3}}{2}}=a.\frac{\sqrt{3}}{3}$$$$2)r_1=r_2=r_3=p.tg\frac{A}{2}=\frac{3a}{2}.\frac{\sqrt{3}}{3}=a\frac{\sqrt{3}}{2}$$$$\Rightarrow \frac{27r^{2}R}{\mathrm{\Pi }{r}_1}=\frac{27\times \frac{a^2}{12}\times a\frac{\sqrt{3}}{3}}{a^3\times \frac{3\sqrt{3}}{8}}=2.◼$$

Commented by Tinkutara last updated on 31/Jul/17

$$\mathrm{Thank}\mathrm{you}\mathrm{very}\mathrm{much}\mathrm{behi}\mathrm{Sir}!$$

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