If the sides a, b, c of a ∆ABC is in H.P., prove that are in H.P.
As a, b, c is in HP (given)
So, are in AP
Hence
Let a, b, c be the sides of any triangle ABC. Then by applying the sine rule, we get
So, a = k sin A, b = k sin B, c = k sin C…(ii)
Substituting equation (ii) in equation (i), we get
By cross multiplying we get
Now, so above equation becomes,
so the above equation becomes
Divide both sides by , we get
Now canceling the like terms we get
Hence are in AP
Therefore,
are in HP
Hence proved