From Perpendicular Center Lines to Concyclic Points

Solution

The problem is easily solved by angle chasing. Observe the presence of several cyclic quadrilaterals, e.g.,

This explains an interesting distribution of equal angles:

Recombining suitable angles we see that the opposite angles at the four points of tangent intersections do indeed add up to $180^{\circ},\,$ making the points concyclic.

Acknowledgment

The problem has been kindly posted by Dao Thanh Oai at the CutTheKnotMath facebook page.