Explanation

(b)
Let $P_{1},\ P_{2},\ P_{3}$ be altitudes from $P,\ Q$ and $R$
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$\therefore P_{1} = c\sin{Q = \lambda bc,\ P_{2} = a\sin{R = \lambda ca,\ P_{3} = b\sin{P = \lambda ab}}}$
$\left\lbrack \because\ \frac{\sin P}{a} = \frac{\sin Q}{b} = \frac{\sin R}{c} = \lambda \right\rbrack$
Since, $P_{1},\ P_{2},\ P_{3}$ are in AP.
$\Rightarrow \lambda bc,\ \lambda ca,\ \lambda ab$ are in AP.
$\Rightarrow bc,\ ca\ ab$ are in AP .
$\Rightarrow \frac{abc}{a},\frac{abc}{b},\frac{abc}{c}$ are in AP.
$\therefore\ a,\ b,\ c$ are in HP.
$ie,$ sides of the triangle are in HP.