Closing Theorems for Circle Chains

We consider closed chains of circles C₁,C₂,…,Cₙ,Cₙ₊₁=C₁ such that two neighbouring circles Cᵢ,Cᵢ₊₁ intersect or touch each other with Aᵢ being a common point. We formulate conditions such that a polygon with vertices Xᵢ on Cᵢ, and Aᵢ on the (extended) side XᵢXᵢ₊₁, is closed for every position of the starting point X₁ on C₁. Similar results apply to open chains of circles. It turns out that the intersection of the sides XᵢXᵢ₊₁ and XⱼXⱼ₊₁ of the polygon lies on a circle Cᵢⱼ through Aᵢ and Aⱼ with the property that Cᵢⱼ, Cⱼₖ and Cₖᵢ pass through a common point. The six circles theorem of Miquel and Steiner's quadrilateral Theorem appear as special cases of the general results.