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  4. The exradii of a triangle, r1,r2,r3 are in 1A armonic progression, then the sides a, b and c are in

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Q. The exradii of a triangle, $ {{r}_{1}},{{r}_{2}},{{r}_{3}} $ are in 1A armonic progression, then the sides a, b and c are in

Bihar CECEBihar CECE 2015

Solution:

It is given that $ {{r}_{1}},{{r}_{2}} $ and $ {{r}_{3}} $ are in HP. $ \Rightarrow $ $ \frac{2}{{{r}_{2}}}=\frac{1}{{{r}_{1}}}+\frac{1}{{{r}_{3}}} $ $ \Rightarrow $ $ \frac{2(s-b)}{\Delta }=\frac{s-a}{\Delta }+\frac{s-c}{\Delta } $ $ \Rightarrow $ $ 2a-2b=2s-a-c $ $ \Rightarrow $ $ 2b=a+c $ Hence, a, b and c are in AP