The exradii `r_1,r_2` and `r_3` of ` /_\A B C` are in H.P. Show that its sides `a , ba n dc` are in `AdotPdot`

Explanation

To show that the sides \( a, b, c \) of triangle \( \triangle ABC \) are in Arithmetic Progression (AP) given that the exradii \( r_1, r_2, r_3 \) are in Harmonic Progression (HP), we can follow these steps:

### Step 1: Understanding the relationship between exradii and sides
We know that the exradii \( r_1, r_2, r_3 \) of triangle \( \triangle ABC \) are given by:
\[ r_1 = \frac{\Delta}{s-a}, \quad r_2 = \frac{\Delta}{s-b}, \quad r_3 = \frac{\Delta}{s-c} \]
where \( \Delta \) is the area of the triangle and \( s \) is the semi-perimeter defined as:
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