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If in a triangle $\mathrm{ABC}$, the altitudes from the vertices $\mathrm{A}, \mathrm{B}, \mathrm{C}$ on opposite sides are in H.P., then $\sin A, \sin B, \sin C$ are in
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A.P.
A.P.
$\Delta=\frac{1}{2} p_1 a=\frac{1}{2} p_2 b=\frac{1}{2} p_3 b$
$p_1, p_2, p_3$ are in H.P.
$\Rightarrow \frac{2 \Delta}{a}, \frac{2 \Delta}{b}, \frac{2 \Delta}{c}$ are in H.P.
$\Rightarrow \frac{1}{a}, \frac{1}{b}, \frac{1}{c}$ are in H.P
$\Rightarrow a, b, c$ are in A.P.
$\Rightarrow \sin A, \sin B, \sin C$ are in A.P.
$p_1, p_2, p_3$ are in H.P.
$\Rightarrow \frac{2 \Delta}{a}, \frac{2 \Delta}{b}, \frac{2 \Delta}{c}$ are in H.P.
$\Rightarrow \frac{1}{a}, \frac{1}{b}, \frac{1}{c}$ are in H.P
$\Rightarrow a, b, c$ are in A.P.
$\Rightarrow \sin A, \sin B, \sin C$ are in A.P.
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