This turned out to be almost disappointingly easy to prove.
AC is equal in length to AD, since they are radii of the same circle. C is on the vertical line that bisects AD. Therefore, CD is the mirror image of AC, and the triangle ACD is equilateral. Therefore, the angle formed by AC and AD is 60 degrees, and that formed by AB and AC is 30 degrees.
Similar triangles can be made for all the other red points, showing that they are located at 30-degree intervals.
No other grids (3-by-3, 5-by-5, etc.) divide a circle evenly like this.
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