C.a.R. > Applications> Similarity and Angles > Reflection at a Circle
This is one way to construct the reflection at a circle. You need to prove that the two angles are equal. Then the triangles MQP* and MPQ are similar. Now we get the equality shown in the construction.
A special case is MQ=1. then MP = 1 / MP*.
Here is another construction, which works in all cases.
Here is still another construction, which does not work always.
The default macro "reflection at circle" uses a computation instead.