In an acute angled ∆ABC, r_2 + r_3 = r + r_1 and then U;wudks.k ∆ABC esa, r_2 + r_3 = r + r_1rFkk

JEE MainMathematicsModerateProperties of triangle

In an acute angled ∆ABC, r_2 + r_3 = r + r_1 and
$\angle B > \frac{\pi}{3}$ then
U;wudks.k ∆ABC esa, r_2 + r_3 = r + r_1rFkk

Show answer and solution

r - r_{2} = r_{3} - r_{1}

\frac{\Delta}{S} - \frac{\Delta}{S - b} = \frac{\Delta}{S - c} - \frac{\Delta}{S - a}

\Rightarrow \frac{- b}{s(s - b)} = \frac{c - a}{(S - a)(s - c)}

\frac{(S - a)(S - c)}{S(S - b)} = \frac{a - c}{b}

\Rightarrow \tan^{2}\frac{B}{2} = \frac{a - c}{b}

But
ijUrq

\frac{B}{2} \in \left( \frac{\pi}{6},\frac{\pi}{4} \right) \Rightarrow \tan^{2}\frac{B}{2} \in \left( \frac{1}{3},1 \right)

\Rightarrow \frac{1}{3} < \frac{a - c}{b} < 1

\Rightarrow b < 3a - 3c < 3b

\Rightarrow b + 3c < 3a < 3b + 3c

Source: Grass-G / M-26 Mathematics Paper 1 15 Jan.docx

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