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Question Number 174779 by mnjuly1970 last updated on 10/Aug/22

Commented by behi834171 last updated on 10/Aug/22

$$\mathit{s}\mathit{i}\mathit{r}!\mathit{y}\mathit{o}\mathit{u}\mathit{r}\mathit{u}\mathit{s}\mathit{e}\mathit{d}\mathit{f}\mathit{o}\mathit{n}\mathit{t}\mathit{s}\mathit{a}\mathit{r}\mathit{e}\mathit{t}\mathit{o}\mathit{o}\mathit{s}\mathit{m}\mathit{a}\mathit{l}\mathit{l}.$$$$\mathit{p}\mathit{l}\mathit{e}\mathit{a}\mathit{s}\mathit{e}\mathit{b}\mathit{e}\mathit{f}\mathit{r}\mathit{i}\mathit{e}\mathit{n}\mathit{d}\mathit{w}\mathit{i}\mathit{t}\mathit{h}\mathit{m}\mathit{y}\mathit{e}\mathit{y}\mathit{e}\mathit{s}\mathit{a}\mathit{n}\mathit{d}$$$$\mathit{u}\mathit{s}\mathit{e}\mathit{l}\mathit{a}\mathit{r}\mathit{g}\mathit{e}\mathit{r}\mathit{f}\mathit{o}\mathit{n}\mathit{t}\mathit{s}.\mathit{t}\mathit{h}\mathit{a}\mathit{n}\mathit{k}\mathit{s}\mathit{i}\mathit{n}\mathit{a}\mathit{d}\mathit{v}\mathit{a}\mathit{n}\mathit{c}\mathit{e}.$$

Commented by mnjuly1970 last updated on 11/Aug/22

$$yessir...thanksalot...$$

Answered by behi834171 last updated on 10/Aug/22

$$\mathit{c}\mathit{o}\mathit{s}\frac{\mathit{A}}{2}=\sqrt{\frac{\mathit{p}(\mathit{p}-\mathit{a})}{\mathit{b}\mathit{c}}}\mathit{a}\mathit{n}\mathit{d}\mathit{s}\mathit{o}\mathit{o}\mathit{n}$$$$\mathrm{\Pi }\mathit{c}\mathit{o}\mathit{s}\frac{\mathit{A}}{2}=\frac{\mathit{p}\sqrt{\mathit{p}(\mathit{p}-\mathit{a})(\mathit{p}-\mathit{b})(\mathit{p}-\mathit{c})}}{\mathit{a}\mathit{b}\mathit{c}}=$$$$=\frac{\mathit{p}.\mathit{S}}{\mathit{a}\mathit{b}\mathit{c}}=\frac{\mathit{p}}{4\mathit{R}}=\frac{\mathit{a}+\mathit{b}+\mathit{c}}{8\mathit{R}}=\frac{2\mathit{R}\mathrm{\Sigma }\mathit{s}\mathit{i}\mathit{n}\mathit{A}}{8\mathit{R}}⩽$$$$⩽\frac{1}{4}\times 3\times \frac{\sqrt{3}}{2}=\frac{3\sqrt{3}}{8}.◼$$$$[\mathit{n}\mathit{o}\mathit{t}\mathit{e}:\mathrm{\Sigma }\mathit{s}\mathit{i}\mathit{n}\mathit{A}⩽3\times \mathit{s}\mathit{i}\mathit{n}\frac{\mathit{\pi }}{3}=\frac{3\sqrt{3}}{2}]$$

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