In \(\Delta \mathrm{PQR}, \sin \mathrm{P}, \sin \mathrm{Q}\) and \(\sin \mathrm{R}\) are in A.P., then

Let \(\mathrm{p}_1, \mathrm{p}_2, \mathrm{p}_3\) be the altitudes of \(\triangle \mathrm{PQR}\)

Area of \(\triangle P Q R\)
\(\begin{aligned} & =\frac{1}{2} \times \text { base } \times \text { height } \\ & =\frac{1}{2} \times \mathrm{p}_1 \times \mathrm{a}\end{aligned}\)
Area \(=\frac{1}{2} \mathrm{p}_1 \mathrm{a}\)
\(\begin{array}{ll}
\therefore & \mathrm{p}_1=2 \times \frac{\text { Area }}{\mathrm{a}} ... (i)\\
\therefore & \text { similarly, } \\
& \mathrm{p}_2=\frac{2 \times \text { Area }}{\mathrm{b}} ... (ii) \\
& \mathrm{p}_3=\frac{2 \times \text { Area }}{\mathrm{c}} ... (iii)
\end{array}\)
\(\therefore\) By sine Rule,
\(\frac{a}{\sin P}=\frac{b}{\sin Q}=\frac{c}{\sin R}\)
Let \(\frac{a}{\sin P}=\frac{b}{\sin Q}=\frac{c}{\sin R}=k\)
\(\therefore \sin P=\frac{a}{k}, \sin Q=\frac{b}{k}, \sin R=\frac{c}{k}\)
\(\sin P, \sin Q\) and \(\sin R\) are in A.P.
\(\therefore \mathrm{a}, \mathrm{b}, \mathrm{c}\) are in A.P.
\(\therefore \frac{1}{a}, \frac{1}{b}, \frac{1}{\mathrm{c}}\) are in H.P. ... (iv)
From equations (i), (ii), (iii) and (iv), we get \(\mathrm{p}_1, \mathrm{p}_2\) and \(\mathrm{p}_3\) are in H.P.