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 X_P. Extend the line X_PQ to meet c at X_Q. Finally, extend the line X_QR to meet c at X_R. Define f(X) = X_R. Do the iterates fⁿ(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 https://www.java.com/en/download/index.jsp, download and install Java VM and enjoy the applet.

What if applet does not run?

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