Iterations on a Circle Through Three Points

Prof. W. McWorter passed me on the following problem:

  Consider a circle c and three points P, Q, and R inside c. For any point X on c, do the following. Extend the line XP to meet c at XP. Extend the line XPQ to meet c at XQ. Finally, extend the line XQR to meet c at XR. Define f(X) = XR. Do the iterates f n(X) converge to a point X' on c such that f(X') = X'?

The applet below helps to experiment with the problem. (The problem remains meaningful even if some or all of the points P, Q, R lie outside the circle.)


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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