The terms a, b, c and d form a geometric progression. So we can write the terms as b = ar , c = ar^2 and d = ar^3

We have to prove (a - d)^2 = (b - c)^2 + (c - a)^2 + (d - b)^2

We start with the left hand side:

(a - d)^2

=> (a - ar^3)^2

=> a^2(1 - r^3)^2 ...(1)

The identity cannot be proved for (a - d)^2 = (b - c)^2 + (c - a)^2 + (d - b)^2, instead it should be (a - d)^2 = (b - c)^2 - (c - a)^2 + (d - b)^2

(b - c)^2 - (c - a)^2 + (d - b)^2

=> ( ar - ar^2)^2 - ( ar^2 - a)^2 + (ar^3 - ar)^2

=> a^2[(1 - r^2)^2 - (r^2 - 1)^2 + ( r^3 - 1)^2]

=> a^2[ 1 + r^4 - 2r^2 - r^4 - 1 + 2r^2 + r^6 + 1 - 2r^3]

=> a^2[ r^6 + 1 - 2r^3]

=> a^2( 1 - r^3 )^2 ...(2)

From (1) and (2) we get (a - d)^2 = (b - c)^2 - (c - a)^2 + (d - b)^2

The identity that can be proved using the terms a, b, c and d of a GP is (a - d)^2 = (b - c)^2 - (c - a)^2 + (d - b)^2