54.

If are in A.P., show that are in A.P.

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are in A.P.

or  

       are in A.P.   (By subtracting (ab + bc + ca) from each term)

or     are in A.P.

or    a(a - b) - c (a - b), -b (a - b) + c (a - b), c (c - a) - b(c - a) are in A.P.

or    (a - b) (a - c), (a - b) (c - b), (c - a) (c - b) are in A.P.

or    - (a - b), (c - a), - (a - b) (b - c), - (c - a) (b - c) are in A.P.

or     are in A.P.    {By Dividing  each number by (a - b) (c - a) (b - c)}

or      are in A.P.           {By multiplying each number by -1}