[As indicated, I found this on rec.puzzles and thought it might be of interest here as well. Please note that I am not the author. -- steve]
Newsgroups: rec.puzzles From: firstname.lastname@example.org (Frank Pinto) Subject: Circles, circles, circles......puzzle. Organization: University of Massachusetts Dartmouth Date: Mon, 22 Nov 1993 18:28:21 GMT
Hello all Here is an interesting one:
Have a good time
Start with a circle O of radius 1 and let a and b be arbitrary real numbers. Construct a first circle of radius a tangent to circle O. Construct a 2nd circle of radius b tangent to O and the a-circle. Construct a 3rd circle of radius b/a tangent to O and the b-circle. Construct a fourth circle of radius 1/a tangent to O and the b/a-circle. Construct a fifth circle of radius 1/b tangent to O and the 1/a-circle. Construct a sixth circle of radius a/b tangent to O and the 1/b-circle.
Note that each new radius is the ratio of the two preceding radii.
Prove that the sixth circle is tangent to the first circle.
--suggested by Bruce Hanson, St. Olaf College -- Frank Pinto (XICO)| The roots of education are bitter, Mathematics, | but the fruit is sweet. Computer Science &| Philosophy | ARISTOTLE