How to use the geometric progression of terms a,b,c,d in identity: (a-d)^2=(b-c)^2+(c-a)^2+(d-b)^2

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