It is said that, are in AP,

Hence,

Taking LCM,

.

Multiply in both denominator and numerator with c in LHS and a in RHS

Since,

a(b + c), b(c + a), c(a + b) are to be proved in A.P.

b(c + a)-a(b + c) = c(a + b)-b(c + a)

bc + ba – ab - ca- = ca + ab – bc - ba

cb - ca = ca - ab

c(b - a) = a(c - b)

Hence, given terms are in AP.