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...XN, BCY0Y1...YN, X0Y0Z0...ZN. The problem above implies that if one of the vertices Xi, i > 0, lies on the line X0Y0, then the vertex Zi lies on AB. Perhaps a little more surprising is the fact that if Xi lies on AB then Zi lies on X0Y0, whereas Xi and Yi are always on the same line: either both are on X0Y0 or both are on AB.

This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at, download and install Java VM and enjoy the applet.

What if applet does not run?

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