Let A be the first term and D be the c.d. of an A.P.
                             ...(i)
                    ...(ii)
                        ...(iii)
L.H.S. = (q - r) a + (r - p)b + (p - q)c = (q - r) [A + (p - 1)D]
                                    + (r - p) [A + (q - 1)D + (p - q) [A + (r - 1)D]
                          (By using values of a, b, c from (i), (ii), (iii) respectively]
         = A(q - r) + (q - r) (p - 1)D + A (r - p) + (r - p) (q - 1)D + A(p - q) + (p - q) (r - 1)D
         = A(q - r + r - p + p - q) + D[Pq - q - rp + r + rq - r - pq + p + pr - p - qr + q]
         = A(0) + D(0) = 0 = R.H.S.