We have
(a + bx) / (a - bx) = (b + cx) / (b - cx)
==> (a + bx) (b - cx) = (a - bx) (b + cx)
==> ab + (b ² - ac) x - bcx ² = ab + (ac - b ²) x - bcx ²
==> (b ² - ac) x = (ac - b ²) x
==> b ² - ac = ac - b ²
==> 2b ² = 2ac
==> b ² = ac … … … [1]
Similarly, you would find
(a + bx) / (a - bx) = (c + dx) / (c - dx)
==> ad = bc … … … [2]
and
(b + cx) / (b - cx) = (c + dx) / (c - dx)
==> c ² = bd … … … [3]
Now:
c ² = bd ==> b = c ² / d
b ² = ac ==> c = b ² / a
ad = bc ==> d = bc / a
and we find
d / c = (bc / a) / (b ² / a) = c / b
and
c / b = (b ² / a) / (c ² / d) = (b ² d) / (a c ²) = b / a
which is to say, the ratio between d and c is equal to the ratio between c and b, and also equal to the ratio between b and a. Therefore (a, b, c, d) are in a geometric progression.
