Overlay the equilateral triangle on your circle of fifths. Notice the notes at each of the three points of the triangle. If we arrange them in ascending order the result is C-E-G#, the notes needed to form an Augmented 5th Chord. You’ll also notice that as the three sides of the equilateral triangle are of equal length, the intervals of the augmented 5th chord are also equal in length. C to E is a major third interval. E to G# is also a major third interval, and if we add in the octave C (C-E-G#-C)G# to C is also a major third interval.
As you’ll remember from a previous post, the equal sides of the square mirror the equal minor third intervals of the diminished chord.
Observation: Equal sides, equal intervals.