If a, b, c are in AP, prove that bc, ca, and ab are in HP.

We have to prove that bc, ca, and ab are in HP, which is another way of saying that 1/bc, 1/ca, and 1/ab are in AP. This simply means we have to prove that

2(1/ca) = 1/ab + 1/bc
=> 2/ca = 1/ab + 1/bc

We are given that a, b, and c are in AP, which means

2b = a + c

Diving the entire expression by abc, we'll have

=> 2b/abc = a/abc + c/abc
=> 2/ca = 1/bc + 1/ab
which means 1/bc, 1/ca, and 1/ab are in AP, or bc, ca and ab are in AP