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Question Number 198182 by Blackpanther last updated on 13/Oct/23

Answered by mr W last updated on 14/Oct/23

Commented by mr W last updated on 14/Oct/23

$$r=radiusofcircles=1$$$$\mathrm{\angle }QPS=60°=\mathrm{\angle }QRP$$$$\mathrm{\angle }PSQ=\mathrm{\angle }QRP/2=30°$$$$\Rightarrow \mathrm{\Delta }PQS=rightangled$$$$\mathrm{\Delta }ABC\sim \mathrm{\Delta }QPS$$$$AB=3r+\sqrt{3}r=(3+\sqrt{3})r$$$$\mathrm{\Delta }ABC=\frac{AB\times AC}{2}$$$$=\frac{{\left(AB\right)}^2\times \sqrt{3}}{2}=\frac{{(3+\sqrt{3})}^2\sqrt{3}{r}^2}{2}$$$$=(9+6\sqrt{3})r^2=9+6\sqrt{3}✓$$

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