Geometric Circle Puzzle

 My initial instinct for this puzzle was to examine it by summing opposite pairs. For example, I wondered if all opposite pairs summed to 31. This is obviously not correct, but I think that my mind went there because some similar geometric puzzles I have solved in the past can be approached by that method. I wanted to record that thinking here because it was an example of how my past experience in similar puzzles had a direct and immediate impact on my approach to problem-solving. 

When I actually sat down to solve the problem, it became much clearer to me how I might go about it. I know that a clock (which has 12 points) has a difference of 6 between each point. This makes sense, since 6 is half of 12 and the opposed points have equal circumference between them in either direction. I can use this knowledge to extend to a larger circle of 30, knowing that opposed points must all be different by 15. To show this more rigorously, we can add in modulo 30 and know that the difference between points a and b must be d, such that 

a + d = b

and

b + d = a

in mod 30. Thus, d is 15 and 7 is opposite 22.

In terms of extensions, I am very curious about how one might use shapes other than circles to ask similar questions. Equilateral triangles are the first that come to mind, marking different places on one side and looking at the rotationally symmetrical points. In that case, it would be crucial that the number of points be a multiple of 3, as opposed to a multiple of 2 in the case of the circle.

I don't have a strict definition of what a geometric puzzle is, but my intuitive understanding is one in which the simplest proof is a visual one. in this case, one could simply draw the given circumstance and arrive at the answer with no algebraic or logical work whatsoever.