What are the main features of the scale triangles
Are there geometric symmetries in musical harmonics?
If the Barrel net is laid out as follows:
we see that these triads are not right triangles, but equilateral triangles.
Although there are symmetries related to diatonic harmony (and diatonic scales etc) - which are interesting and worth discussing! -, the symmetries you show and ask for are really just a by-product of your choice around yours To lay out the barrel net .
In this regard, note that the consonant triads - major and minor - both consist of a small third and a large third. (The major triad has the major third on the bottom and the minor third on top, and the minor triad reverses this.) In addition, the remaining interval can be thought of as the perfect quarter.
The small third is made up of 3 semitones, the big third is made up of 4, and the perfect fourth is made up of 5. So if you get your geometric space right, it's pretty easy to walk away with a 3: 4: 5 triangle.
Is this a known property that there are right triangles in triadic structures of wave harmonics?
No. I am drawing on your use of (right) triangles and the idea of the hypotenuse, which implies sizes that are in quadrature ala add the Pythagorean theorem.
Our sense of tone is closely related to sounds, which are made up (good approximations of) of an overtone series that are integral multiples of a fundamental, and our sense of harmony relates only to these types of sounds. No quadrature is added here, so no triangles.
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