Three Similar Polygons

The following problem and its generalization have been discussed elsewhere:

Suppose A, B, C are arbitrary points on a straight line and X is a point not on the line. Construct similar and similarly oriented triangles ABX and BCY. If triangle XYZ is similar to triangles ABX and BCY but with a different orientation then Z is always collinear with A, B, and C!

The applet below presents an additional twist. Consider similar polygons: ABX₀X₁...X_N, BCY₀Y₁...Y_N, X₀Y₀Z₀...Z_N. The problem above implies that if one of the vertices X_i, i > 0, lies on the line X₀Y₀, then the vertex Z_i lies on AB. Perhaps a little more surprising is the fact that if X_i lies on AB then Z_i lies on X₀Y₀, whereas X_i and Y_i are always on the same line: either both are on X₀Y₀ or both are on AB.

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