Properties of Triangles 3 Question 19
19. The exradii $r _1, r _2, r _3$ of $\triangle A B C$ are in HP, show that its sides $a, b, c$ are in AP.
$(1983,3 M)$
Integer Answer Type Question
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Solution:
- Since, $r _1, r _2$ and $r _3$ are exradii of $\triangle A B C$ are in HP.
$$ \begin{array}{ll} \therefore & \frac{1}{r _1}, \frac{1}{r _2}, \frac{1}{r _3} \text { are in AP. } \\ \Rightarrow & \frac{s-a}{\Delta}, \frac{s-b}{\Delta}, \frac{s-c}{\Delta} \text { are in } AP . \\ \Rightarrow & s-a, s-b, s-c \text { are in } AP . \\ \Rightarrow & -a,-b,-c \text { are in } AP . \\ \Rightarrow & a, b, c \text { are in AP. } \end{array} $$
