The classical configuration connected with the names of Miquel and Clifford associates with a set of n straight lines in a plane a point called their Miquel point when n is even, and a circle, their Clifford circle, when n is odd. When the number of lines n in a set is even, the omission of these, one at a time, gives n subsets whose associated Clifford circles are concurrent at the Miquel point of the original set; when n is odd, the omission of the lines one at a time gives n subsets whose associated Miquel points are concyclic on the Clifford circle of the original set. To start the chain, it is only necessary to define the Miquel point of two lines as their common point.