Hans Walser, 
Collinear and cocyclic points
Let a, b be parallel lines, and c a circle between a and b touching a in A (Fig. 1a), and d a circle between a and b touching b in B and c in C (Fig. 1b).
Fig. 1: Circles between two parallel lines
The three points A, B, C are collinear (Fig. 2a).
Fig. 2: Collinear points. Proof without words
The figure 2b gives a proof without words.
We reflect the figure 2a about an arbitrary circle (purple in Fig. 3).
Fig. 3: Reflection about a circle
Thus we get a sequence of touching circles. The touching points are cocyclic (Fig. 4).
Fig. 4: Cocyclic points