If in a triangle $P Q R$ , $sin P, sin Q, sin R$ are in $A . P .$ , then prove that the altitudes are in $H . P .$

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Solution

Altitude $P D$ , $Q E$ and $R F$ are $q sin R$ , $r sin P$ , $p sin Q$ where $p, q, r$ are the sides
Apply sine rule.
or $K sin Q sin R, k sin R sin P, k sin p sin Q$
or $\frac{k sin P sin Q, sin R}{sin P}, \frac{k sin P sin Q sin R}{sin Q}, \frac{k sin P sin Q sin R}{sin R}$
But $sin P, sin Q, sin R$ are in $A . P .$
$∴$ Altitudes are in $H . P .$

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