Notes in C major scale are:
C D E F G A B C
If you will treat each note like a root of possible chord and use only notes form scale it will looks like:
C-E-G (Cmaj - I)
D-F-A (Dmin - ii)
E-G-B (Emin - iii)
F-A-C (Fmaj - IV)
G-B-D (Gmaj - V)
A-C-E (Amin - vi)
B-D-F (Bdim - vii dim)
(simply take every second note)
Now check the relation between notes in each chord. They will be exactly like those described by you in your first post :)