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Question Number 145573 by mnjuly1970 last updated on 06/Jul/21

Answered by Olaf_Thorendsen last updated on 06/Jul/21

$$p=halfperimeter$$$$p=\frac{1}{2}(1+2r+2+2r+3+2r)=3(r+1)$$$$p-a=3(r+1)-(1+2r)=r+2$$$$p-b=3(r+1)-(2+2r)=r+1$$$$p-c=3(r+1)-(3+2r)=r$$$$S=\sqrt{p(p-a)(p-b)(p-c)}$$$$S=\sqrt{3(r+1)(r+2)(r+1)r}$$$$S=(r+1)\sqrt{3r(r+2)}$$$$$$$$S_1=\frac{1}{2}{\theta }_1{r}^2$$$$S_2=\frac{1}{2}{\theta }_2{r}^2$$$$S_3=\frac{1}{2}{\theta }_3{r}^2$$$$S_1+S_2+S_3=\frac{1}{2}({\theta }_1+{\theta }_2+{\theta }_3)r^2=\frac{1}{2}\pi r^2$$$$\frac{S_1+S_2+S_3}{S}=\frac{\frac{1}{2}\pi r^2}{(r+1)\sqrt{3r(r+2)}}=\frac{\pi }{6}$$$$(r+1)\sqrt{3r(r+2)}=3r^2$$$${(r+1)}^{2}3r(r+2)=9r^4$$$${(r+1)}^2(r+2)=3r^3$$$$(r^2+2r+1)(r+2)=3r^3$$$$-2r^3+4r^2+5r+2=0$$$$r\approx 2,959$$

Commented by mnjuly1970 last updated on 06/Jul/21

$$thanksalotmrolaf$$

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