# 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: ABX_{0}X_{1}...X_{N}, BCY_{0}Y_{1}...Y_{N}, X_{0}Y_{0}Z_{0}...Z_{N}. The problem above implies that if one of the vertices X_{i}, _{0}Y_{0}, 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 X_{0}Y_{0}, whereas X_{i} and Y_{i} are always on the same line: either both are on X_{0}Y_{0} or both are on AB.

What if applet does not run? |

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