Possible roots lenghts in an irreducible root system

[Sarosh]

If a root system R is irreducible (not a product of two root systems) then R does not contain three vectors of pairwise different lengths.

To show this do we need just to compute all the angles and the ratios between the lenghts of roots as explained (1) in the following post:

Two questions on roots of finite, simple, complex lie algebra

Howeber how the fact that R is irreducible is used/involved?
Can you suggest another method or approach to show the claim.