Prove that a,b,c are in A.P. iff 1/bc,1/ca,1/ab are also in A.P.
Consider 1/bc, 1/ca, 1/ab are in AP,
So, 1/ca = (1/bc + 1/ab)/2
or, 1/ca = (a+c)/2abc
or, 1/ca = (a+c)/2abc
or, 1 = (a+c)/2b
or, b = (a+c)/2
Thus, 'b' is A.M. of 'a' and 'c',
∴ a, b, c are in A.P. if 1/bc, 1/ca, 1/ab are in A.P.
Proved