Question

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

• A. the altitudes are in A.P
• B. the altitudes are in H.P
• C. the medians are in G.P
• D. the medians are in A.P

Answer 1

Answer: (B)

the altitudes are in H.P

Let $P,Q,R$ be the triangle with an altitude $PD$ perpendicular to $QR$, altitude $QE$ perpendicular to $PR$, and altitude $RF$ perpendicular to $PQ$.
Therefore, $l\left(QR\right)=p$, $l\left(PR\right)=q$, and $l\left(PQ\right)=r$.
Also, $\sin P,\sin Q,\sin R$ are in A.P.
Therefore, Altitudes $PD,QE,$ and $RF$ will be $q\sin R,r\sin P,$ and $p\sin Q.$
Now, applying \sine rule, we get
$k\sin Q\sin R,k\sin R\sin P,k\sin P\sin Q$
or $\frac{k\sin P\times \sin Q\times \sin R}{\sin P},\frac{k\sin P\times \sin Q\times \sin R}{\sin Q},\frac{k\sin P\times \sin Q\times \sin R}{\sin R}$
But we know that, $\sin P,\sin Q,\sin R$ are in A.P.
$\Rightarrow$ Altitudes are in H.P.