In a triangle ABC- r1-ra-r2-r b-r3-r c-

We know, r_1, r_2 ,r_3 are the three exradii respectively. ⇒r_1 ra+r_2 r b+r_3 r c ⇒1a [ Δs a Δs ]+1b[Δs b Δs]+1c[Δs c Δs] ⇒Δa [ s (s a)s(s a) ]+Δb[s (s b)s(s b ...

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Standard XII

Mathematics

Ex Radius

In a triangle...

Question

In a triangle $A B C, \frac{r_{1} - r}{a} + \frac{r_{2} - r}{b} + \frac{r_{3} - r}{c} =$

\frac{r_{1} + r_{2} + r_{3}}{3}

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r_{1} + r_{2} + r_{3}

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\frac{r_{1} + r_{2} + r_{3}}{2}

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\frac{r_{1} + r_{2} + r_{3}}{s}

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Solution

The correct option is D $\frac{r_{1} + r_{2} + r_{3}}{s}$
We know, $r_{1}, r_{2}, r_{3}$ are the three exradii respectively.
$\Rightarrow \frac{r_{1} - r}{a} + \frac{r_{2} - r}{b} + \frac{r_{3} - r}{c}$
$\Rightarrow \frac{1}{a} [\frac{Δ}{s - a} - \frac{Δ}{s}] + \frac{1}{b} [\frac{Δ}{s - b} - \frac{Δ}{s}] + \frac{1}{c} [\frac{Δ}{s - c} - \frac{Δ}{s}]$
$\Rightarrow \frac{Δ}{a} [\frac{s - (s - a)}{s (s - a)}] + \frac{Δ}{b} [\frac{s - (s - b)}{s (s - b)}] + \frac{Δ}{c} [\frac{s - (s - c)}{s (s - c)}]$
$\Rightarrow \frac{Δ}{s (s - a)} + \frac{Δ}{s (s - b)} + \frac{Δ}{s (s - c)}$
$\Rightarrow \frac{1}{s} [r_{1} + r_{2} + r_{3}]$
Therefore,
$\Rightarrow \frac{r_{1} - r}{a} + \frac{r_{2} - r}{b} + \frac{r_{3} - r}{c} = \frac{r_{1} + r_{2} + r_{3}}{s}$

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Ex Radius

Standard XII Mathematics

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In a triangle ABC, r1−ra+r2−rb+r3−rc=

In a triangle $A B C, \frac{r_{1} - r}{a} + \frac{r_{2} - r}{b} + \frac{r_{3} - r}{c} =$