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If the reciprocals of the lengths of the sides of a $\triangle A B C$ are in harmonic progression, then its ex-radii $r_1, r_2, r_3$ are in
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Harmonic progression
Let the sides of a $\triangle A B C$ are $a, b$ and $c$
$\because \frac{1}{a}, \frac{1}{b}, \frac{1}{c}$ are in H.P. $\Rightarrow a, b, c$ are in A.P.
$\Rightarrow s-a, s-b, s-c$ are in A.P.
$\Rightarrow \frac{s-a}{\Delta}, \frac{s-b}{\Delta}, \frac{s-c}{\Delta}$ are in A.P.
$\therefore r_1, r_2, r_3$ are in H.P.
$\because \frac{1}{a}, \frac{1}{b}, \frac{1}{c}$ are in H.P. $\Rightarrow a, b, c$ are in A.P.
$\Rightarrow s-a, s-b, s-c$ are in A.P.
$\Rightarrow \frac{s-a}{\Delta}, \frac{s-b}{\Delta}, \frac{s-c}{\Delta}$ are in A.P.
$\therefore r_1, r_2, r_3$ are in H.P.
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