Question

if a,b,c,d are in G.P., show that (b-c)^2+(c-a)^2+(d-b)^2=(a-d)^2

Answer 1

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

Let a = p, b = pr , c = pr² and d = pr³

Now, LHS = (b - c)² + (c - a)² + (d - b)²
= (pr - pr²)² + (pr² - p)² + (pr³ - pr)²
=( p²r² + p²r⁴ - 2p²r³) + (p²r⁴ + p² - 2p²r²) + (p²r^6 + p²r² -2p²r⁴)
= p²r^6 - 2p²r³ + p²
= (pr³) -2(pr³)(p) + p²
= (pr³ - p)²
= (p - pr³)²
=(a - d)² = RHS

Answer 2

Hey there!

a × d = b × c 

Also 
ac = b²
bd = c²

(b-c)² + (c-a)² + (d-b)² = b² + c² - 2 bc + c² + a² - 2 ac + d² + b² - 2 db 

= a² + 2 b² + 2 c² + d² - 2 bc - 2 ac - 2 db 

= a² + 2 b² + 2 c² + d² - 2 ad - 2 ac - 2 db 

= a² - 2 ad + d² + 2 b² + 2 c² - 2 ac - 2 db 

= a² - 2 ad + d² + 2 b² + 2 c² - 2 b² - 2 c² 

= (a - d)²

HOPE IT HELPED ^_^