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Question Number 25602 by behi.8.3.4.17@gmail.com last updated on 11/Dec/17

Commented by behi.8.3.4.17@gmail.com last updated on 11/Dec/17

$$\mathit{A}\mathit{c}\mathit{i}\mathit{r}\mathit{c}\mathit{l}\mathit{e}\mathit{p}\mathit{a}\mathit{s}\mathit{s}\mathit{e}\mathit{s}\mathit{t}\mathit{h}\mathit{r}\mathit{o}\mathit{u}\mathit{g}\mathit{h}$$$$\mathit{m}\mathit{i}\mathit{d}\mathit{p}\mathit{o}\mathit{i}\mathit{n}\mathit{t}\mathit{s}\mathit{o}\mathit{f}\mathit{s}\mathit{i}\mathit{d}\mathit{e}\mathit{s}\mathit{o}\mathit{f}$$$$\mathit{t}\mathit{r}\mathit{i}\mathit{a}\mathit{n}\mathit{g}\mathit{l}\mathit{e}\mathit{A}\stackrel{△}{\mathit{B}\mathit{C}}.$$$$\mathrm{find}:$$$$1)\mathrm{its}\mathrm{radius}\mathrm{in}\mathrm{term}\mathrm{of}:\mathit{a},\mathit{b},\mathit{c}.$$$$2)\mathrm{distance}\mathrm{of}\mathrm{its}\mathrm{center}\mathrm{from}:\mathrm{A},\mathrm{B},\mathrm{C}.$$$$3)\mathrm{distance}\mathrm{from}\mathrm{its}\mathrm{centre}\mathrm{to}$$$$\mathrm{centerpoint}\mathrm{of}\mathrm{out}\mathrm{circle}\mathrm{of}:\mathrm{A}\stackrel{△}{\mathrm{B}C}.$$

Answered by mrW1 last updated on 12/Dec/17

$$DE=\frac{a}{2}=a^{\prime }$$$$DF=\frac{b}{2}=b^{\prime }$$$$EF=\frac{c}{2}=c^{\prime }$$$$r=\frac{a^{\prime }{b}^{\prime }{c}^{\prime }}{4A_{DEF}}=\frac{a^{\prime }{b}^{\prime }{c}^{\prime }}{4\sqrt{s^{\prime }(s^{\prime }-a^{\prime })(s^{\prime }-b^{\prime })(s^{\prime }-c^{\prime })}}$$$$s^{\prime }=\frac{a^{\prime }+b^{\prime }+c^{\prime }}{2}=\frac{a+b+c}{4}=\frac{s}{2}$$$$\Rightarrow r=\frac{abc}{8\sqrt{s(s-a)(s-b)(s-c)}}$$

Commented by behi.8.3.4.17@gmail.com last updated on 12/Dec/17

$$\mathrm{thank}\mathrm{you}\mathrm{so}\mathrm{much}\mathrm{dear}\mathrm{MrW}1.$$$$\text{You can't use 'macro parameter character \#' in math mode}$$$$\mathrm{please}!$$

Commented by mrW1 last updated on 12/Dec/17

$$toQ3:$$$$thecircleaboveisthesocallednine-$$$$pointcircle.itishalfsolargeasthe$$$$circumcircle:r=\frac{R}{2}with$$$$R=radiusofcircumcircleand$$$$R=\frac{abc}{4\sqrt{s(s-a)(s-b)(s-c)}}$$$$Thedistancebetweenthenine-point$$$$circlecenterandthecircumcenter$$$$is$$$$\mathrm{\Delta }L=\frac{1}{2}\sqrt{9R^2-a^2-b^2-c^2}$$

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