Let the side lengths be n-1, n, and n+1. Let the smallest angle be x.
The smallest angle is opposite n-1 and the largest opposite n+1.
From the sine rule we get
n-1 n n+1
-------- = ------------- = ---------
sin(x) sin(180-3x) sin(2x)
n-1 n n+1
==> -------- = --------- = ---------
sin(x) sin(3x) sin(2x)
n-1 n n+1
==> -------- = -------------------- = ---------------
sin(x) sin(x)[4cos2(x)-1] 2sin(x)cos(x)
n-1 n n+1
==> -------- = ------------ = ---------
1 4cos2(x)-1 2cos(x)
n+1 2n-1
==> cos(x) = -------- and cos2(x) = --------
2(n-1) 4(n-1)
==> n(n-5) = 0
Therefore, the side lengths are 4, 5, and 6.
See "complete solution by Daniel for a solution using the cosine rule.
See "Geometric solution!" by Chesca for a solution using similar triangles.
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