If in a triangle PQR, sin P, sin Q, sin R are in A.P, then :

The altitudes are in H.P.

Altitudes PD, QE, RF are q sin R, r sin P, p sin Q, respectively

Where p, q, r are sides of the triangle

Apply sine rule : $\frac{q}{\sin Q}=\frac{r}{\sin R}=\frac{p}{\sin P}=K$

Attitudes are

⇒ K sin Q sin R, K sin R sin P, K sin P sin Q

$⇒\frac{K\sin P\sin Q\sin R}{\sin P}, \frac{K\sin P\sin Q\sin R}{\sin Q}, \frac{K\sin P\sin Q\sin R}{\sin R}$

Given, sin P, sin Q, sin R are in A.P.

⇒ Altitudes are in H.P.