Question and Answers Forum

All Questions      Topic List

Algebra Questions

Previous in All Question      Next in All Question      

Previous in Algebra      Next in Algebra      

Question Number 155130 by mathdanisur last updated on 25/Sep/21

Answered by mr W last updated on 26/Sep/21

Commented by mathdanisur last updated on 26/Sep/21

$$\mathrm{Thank}\mathrm{you}\mathbf{S}\mathrm{er},\mathrm{how}\mathrm{proved}\mathrm{please}$$

Commented by mr W last updated on 26/Sep/21

$$wehavefortriangle\mathrm{\Delta }ABC:$$$$\mathrm{\Delta }=\frac{abc}{4R}=\frac{(a+b+c)r}{2}\Rightarrow \frac{abc}{a+b+c}=2Rr$$$$\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}=2R$$$$4\cos \frac{A}{2}\cos \frac{B}{2}\cos \frac{C}{2}=\sin A+\sin B+\sin C$$$$$$$$intriangle\mathrm{\Delta }BIC:$$$$\frac{a}{\sin \mathrm{\angle }BIC}=2R_a$$$$R_a=\frac{a}{2\sin \mathrm{\angle }BIC}$$$$R_a=\frac{a}{2\sin (\pi -\frac{B}{2}-\frac{C}{2})}$$$$R_a=\frac{a}{2\sin (\pi -\frac{\pi -A}{2})}$$$$R_a=\frac{a}{2\cos \frac{A}{2}}$$$$similarly$$$$R_b=\frac{b}{2\cos \frac{B}{2}}$$$$R_c=\frac{c}{2\cos \frac{C}{2}}$$$$R_a{R}_b{R}_c=\frac{abc}{8\cos \frac{A}{2}\cos \frac{B}{2}\cos \frac{C}{2}}$$$$R_a{R}_b{R}_c=\frac{abc}{2(\sin A+\sin B+\sin C)}$$$$R_a{R}_b{R}_c=\frac{abc}{2(\frac{a}{2R}+\frac{b}{2R}+\frac{c}{2R})}$$$$R_a{R}_b{R}_c=\frac{abcR}{(a+b+c)}=2RrR$$$$\Rightarrow R_a{R}_b{R}_c=2R^{2}r$$

Commented by mathdanisur last updated on 26/Sep/21

$$\mathrm{awesome}\mathrm{solution},\mathrm{thank}\mathrm{you}\mathbf{S}\mathrm{er}$$

Commented by Tawa11 last updated on 26/Sep/21

$$\mathrm{great}\mathrm{sir}$$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com