(A) Given that $r_1, r_2, r_3$ are in $HP$. Since $r_1 = \frac{\Delta}{s-a}$,$r_2 = \frac{\Delta}{s-b}$,and $r_3 = \frac{\Delta}{s-c}$,where $\Delta$

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(A) Given that $r_1, r_2, r_3$ are in $HP$. Since $r_1 = \frac{\Delta}{s-a}$,$r_2 = \frac{\Delta}{s-b}$,and $r_3 = \frac{\Delta}{s-c}$,where $\Delta$ is the area and $s$ is the semi-perimeter. For $r_1, r_2, r_3$ to be in $HP$,we have: $\frac{2}{r_2} = \frac{1}{r_1} + \frac{1}{r_3}$ Substituting the values: $\frac{2(s-b)}{\Delta} = \frac{s-a}{\Delta} + \frac{s-c}{\Delta}$ $2s - 2b = s - a + s - c$ $2s - 2b = 2s - (a + c)$ $-2b = -(a + c)$ $2b = a + c$ This implies that $a, b$ and $c$ are in $AP$.

Class 11 Mathematics Trigonometrical Equations Mix Examples-Trigonometrical Equations and Inequations, Properties of Triangles, Height and Distance

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If $r_1, r_2$ and $r_3$ of a $\triangle ABC$ are in Harmonic progression,then $a, b$ and $c$ will be in

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arithmetic progression
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geometric progression
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