I am trying to understand the proof of Copernicus' Theorem.
Consider two circles of radii R and R/2 with the smaller one rolling inside the bigger circle without slipping. Copernicus' Theorem states a surprising result that a point on the circumference of the small circle traces a straight line segment - a diameter of the big circle, to be precise.
If found the following useful resource with a proof.
I was able to understand everything of the proof besides the first argument:
Assume point M on the small circle has previously occupied the position of point N on the large circle. Since there is no slipping, arcs PM (on the small circle) and PN (on the large circle) have exactly the same length.
I can't conclude that the arcs of PM and PN (marked as red) should have the same lenght. Is there a proof for that?