Reflection of a circle Someone submitted this problem to me:

A physicist would argue that light is a wave and that it passes anyway.
However here we are dealing with a problem of pure geometry.
My first instinct was to try and find a way that it does follow some kind of easy rule, but it did not work.
My second attempt was to try and find a pattern when solving analytically the problem for low values of N reflections. Here is the Desmos simulation of my attempt: https://www.desmos.com/calculator/k9cuxxn4x5
It is very basic geometry with tangent calculation (calculation of lines intercepting a circle, calculation of tangent and calculation of reflexions), and the analytic geometry becomes complicated for large N, so I assume there is an easy way that I completely missed.
In this Desmos sheet I called a the distance between the top of the circle and the mirror.
Ray component on x-axis is always to the right. 

When gap is zero, angles of incidence are progressively diminishing to converge towards zero  monotonously. Reflected rays are confined to the left of arbelos area and undergo infinitely many bounces but never leave the zero gap tangency point.

A positive gap allows a positive rightward horizontal component by laws of reflection and the rays converge before diverging. Reflected angles at fist reduce and then increase. That is, rays eventually escape. A rough typical ray-trace is sketched having forward components.
Ray component on x-axis is always to the right. 

When gap is zero, angles of incidence are progressively diminishing to converge towards zero  monotonously. Reflected rays are confined to the left of arbelos area and undergo infinitely many bounces but never leave the zero gap tangency point.

A positive gap allows a positive rightward horizontal component by laws of reflection and the rays converge before diverging. Reflected angles at fist reduce and then increase. That is, rays eventually escape. A rough typical ray-trace is sketched having forward components.