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Question Number 35167 by behi83417@gmail.com last updated on 16/May/18

Commented by behi83417@gmail.com last updated on 16/May/18

$$\text{You can't use 'macro parameter character \#' in math mode}$$$$AM=AN,sin\frac{A}{2}=\frac{MN}{2AN}\Rightarrow MN=2AN.sin\frac{A}{2}$$$$AMON,iscyclic,so:$$$$AO.MN=AM.ON+AN.OM\Rightarrow$$$$2r.AN=AO.2AN.sin\frac{A}{2}\Rightarrow AO=\frac{r}{sin\frac{A}{2}}$$$$AD=AO+OD=\frac{r}{sin\frac{A}{2}}+r=r(1+\frac{1}{sin\frac{A}{2}})$$$$\frac{DC}{sin\frac{A}{2}}=\frac{AC}{sinB}\Rightarrow DC=\frac{b}{sinB}sin\frac{A}{2}=2R.sin\frac{A}{2}$$$$\Rightarrow DC=BD=2R.sin\frac{A}{2}$$$$ABDCiscyclic.so:$$$$AD.BC=AB.DC+AC.BD\Rightarrow$$$$a.r.(1+\frac{1}{sin\frac{A}{2}})=c.2R.sin\frac{A}{2}+b.2R.sin\frac{A}{2}$$$$\Rightarrow r(1+sin\frac{A}{2})=2R\frac{b+c}{a}.{sin}^2\frac{A}{2}\Rightarrow$$$$\Rightarrow \mathit{r}=2\mathit{R}.\frac{\mathit{b}+\mathit{c}}{\mathit{a}}.\frac{{\mathit{s}\mathit{i}\mathit{n}}^2\frac{\mathit{A}}{2}}{1+\mathit{s}\mathit{i}\mathit{n}\frac{\mathit{A}}{2}}.◼$$$$note:$$$$R=\frac{abc}{4S_{A\stackrel{△}{BC}}},sin\frac{A}{2}=\sqrt{\frac{(p-b)(p-c)}{bc}},p=\frac{a+b+c}{2}$$

Commented by ajfour last updated on 16/May/18

$$AODmaynotbestraight,Sir.$$

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