The red line AE is the path of a particle which broke free from a rotating disc at A. It's subsequent positions at equal intervals of time on the disc are marked by black dots. The dotted blue circle is the path followed by the point from where the particle broke off. Notice that AB, BC, CD and DE are equal in length and angles AOF, FOG, GOH and HOI are equal.
Relative to the disc A, F, G, H and I are stationary points and the green lines define the positions of the moving particle at equal intervals of time. The green lines are seen to rotate and get longer.