Question

In triangle ABC, $\sin A,\sin B,\sin C$ are in A.P, then

• A. the altitudes are in H.P.
• B. the altitudes are in G.P.
• C. the altitudes are in A.P.
• D. none of these

Answer 1

The correct option is A the altitudes are in H.P.

Using sine rule $a=k\sin A,b=k\sin B,c=k\sin C$
Now Altitudes $P_1=\frac{2\Delta }{c}=\frac{2\Delta }{k\sin C}$
similarly $P_2=\frac{2\Delta }{b}=\frac{2\Delta }{k\sin B}$
and $P_3=\frac{2\Delta }{a}=\frac{2\Delta }{k\sin A}$
Given $2\sin B=\sin A+\sin C$
$\Rightarrow \frac{2\Delta }{k{P}_2}=\frac{\Delta }{k{P}_1}+\frac{\Delta }{k{P}_3}$
$\Rightarrow \frac{2}{P_2}=\frac{1}{P_1}+\frac{1}{P_3}$
$\therefore P_1,P_2,P_3\in H.P$