2-nd-part-of-Q-16214-Prove-that-r-1-s-tan-A-2-r-2-s-tan-B-2-r-3-s-tan-C-2-

Question Number 16269 by Tinkutara last updated on 20/Jun/17

2^{nd} part of Q . 16214 : Prove that

r_{1} = s \tan (\frac{A}{2}), r_{2} = s \tan (\frac{B}{2}),

r_{3} = s \tan (\frac{C}{2}) .

Answered by mrW1 last updated on 20/Jun/17

Commented by mrW1 last updated on 20/Jun/17

with usage of diagram above from

mr . ajfour we have

AE = AB + BD

AF = AC + CD

r_{1} = AE \times \tan \frac{A}{2}

r_{1} = AF \times \tan \frac{A}{2}

\Rightarrow {2 r}_{1} = (AE + AF) \times \tan \frac{A}{2}

\Rightarrow {2 r}_{1} = (AB + BC + AC) \times \tan \frac{A}{2}

\Rightarrow {2 r}_{1} = 2 s \times \tan \frac{A}{2}

\Rightarrow \tan \frac{A}{2} = \frac{r_{1}}{s}

similarly

\Rightarrow \tan \frac{B}{2} = \frac{r_{2}}{s}

\Rightarrow \tan \frac{C}{2} = \frac{r_{3}}{s}

Commented by Tinkutara last updated on 20/Jun/17

Thanks Sir!