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Question Number 188320 by normans last updated on 27/Feb/23

$$$$$$\mathit{i}\mathit{f}\mathit{t}\mathit{h}\mathit{e}\mathit{o}\mathit{n}△\mathit{A}\mathit{B}\mathit{C},$$$$\mathit{i}\mathit{n}\mathit{n}\mathit{e}\mathit{r}\mathit{c}\mathit{i}\mathit{r}\mathit{c}\mathit{l}\mathit{e}\mathit{r}\mathit{a}\mathit{d}\mathit{i}\mathit{u}\mathit{s}\mathit{a}\mathit{n}\mathit{d}\mathit{o}\mathit{u}\mathit{t}\mathit{e}\mathit{r}\mathit{c}\mathit{i}\mathit{r}\mathit{c}\mathit{l}\mathit{e}\mathit{r}\mathit{a}\mathit{d}\mathit{i}\mathit{u}\mathit{s}\mathit{r}\mathit{a}\mathit{n}\mathit{d}\mathit{R}$$$$\mathit{s}\mathit{h}\mathit{o}\mathit{w}\mathit{t}\mathit{h}\mathit{a}\mathit{t};$$$$\mathit{r}=4\mathit{R}\mathit{s}\mathit{i}\mathit{n}\frac{\mathit{A}}{2}\cdot \mathit{s}\mathit{i}\mathit{n}\frac{\mathit{B}}{2}\cdot \mathit{s}\mathit{i}\mathit{n}\frac{\mathit{C}}{2}$$$$$$

Answered by mr W last updated on 28/Feb/23

$$\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}=2R$$$$\mathrm{\Delta }=\frac{abc}{4R}=2R^2\sin A\sin B\sin C$$$$r=\frac{2\mathrm{\Delta }}{a+b+c}$$$$=\frac{2\mathrm{\Delta }}{2R(\sin A+\sin B+\sin C)}$$$$=\frac{2R^2\sin A\sin B\sin C}{R(\sin A+\sin B+\sin C)}$$$$=\frac{2R\sin A\sin B\sin C}{\sin A+\sin B+\sin C}$$$$=\frac{2R\sin A\sin B\sin C}{4\cos \frac{A}{2}\cos \frac{B}{2}\cos \frac{C}{2}}$$$$=\frac{16R\sin \frac{A}{2}\cos \frac{A}{2}\sin \frac{B}{2}\cos \frac{B}{2}\sin \frac{C}{2}\cos \frac{C}{2}}{4\cos \frac{A}{2}\cos \frac{B}{2}\cos \frac{C}{2}}$$$$=4R\sin \frac{A}{2}\sin \frac{B}{2}\sin \frac{C}{2}$$

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