Reflecting Angles

Within a circular mirror, a ray of light is shown. It hits the mirror at N-1 points before returning to its starting point (at point N). If the ray never retraces its path, determine the possible values for the angle between the incident and reflected rays at each point.

Source: Original.

Solutions were received from Samantha Levin, Parthiv Shah, Gerry Nagel, Joseph DeVincentis, Gary Koerzendorfer, Philippe Fondanaiche, and Kristine Anderson.

Gerry Nagel's solution:
For all positive integers n <= N/2, & n,N are coprime, 180-(n*360/N) degrees. Since no n exists when N=0 & N=1, no solutions exist. 

 
    N  n  angle
    -----------
    2  1  0
    3  1  60
    4  1  90
    5  1  108
    5  2  36
    6  1  120
    7  1  128.57
    7  2  77.14
    7  3  25.71
    8  1  135
    8  3  45
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