Let ABC be the triangle such that the lengths of its sides CA, AB and BC are x – 1, x and x + 1 respectively where x ∈ N and x > 1.
Let ∠B = α be the smallest angle and ∠A = 2α be the largest angle.
Then, by sine rule, we have
using cosine law
⇒ cos α = (x + 4)/2(x - 1)
From Eqs. (i) and (ii), we get
(x + 1)/2(x - 1) = (x + 4)8/2(x + 1)
⇒ (x + 1)2 = (x + 4) (x – 1)
⇒ x2 + 2x + 1 = x2 + 3x – 4⇒ x = 5
Hence, the lengths of the sides of the triangle are 4, 5 and 6 unit.