Hans Walser, [20170708]

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