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.)
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