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Area between Two Curves

Show that r...

Question

Show that ( $r_{2}$ + $r_{3}$ ) / (1 + cos A) = ( $r_{3}$ + $r_{1}$ ) / (1 + cos B) = ( $r_{1}$ + $r_{2}$ ) / (1 + cos C).

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Solution

$1 + c o s A$ = $2 c o s^{2} \frac{A}{2} = 2 \frac{s (s - a)}{b c} = D^{r}$
$r_{2} + r_{3} = \frac{S}{s - b} + \frac{S}{s - c} = \frac{S \cdot a}{(s - b) (s - c)} = N^{r}$
$∴ T_{1} = \frac{N^{r}}{D^{r}} = \frac{S}{S^{2}} \frac{a (b c)}{2} = \frac{a b c}{2 S}$
Hence, $T_{1} = T_{2} = T_{3} = \frac{a b c}{2 S}$ by symmetry.

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