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Math Horizons - November 2003

Content Teasers for November 2003

Paintings, Tilings, and Proofs

The proof is in the paving.

Knots to You

Untangling the wild growth in knot theory.

Math in the ICU

The calculus of cardiac care.

Thou Shalt Not Divide By Zero!

But why not, really?

Circles: Around, About, Across, & Through

Circum-scribing the circle.

Area & Arc Length of Trochogonal Arches

Spinning a polygonal tale.

Problem Section


Proposed by Bill Livingston, Missouri Southern State University. The product of three real numbers is -1/2 and the sum of their squares is 57/4. If the three numbers are taken two at a time and multiplied, the sum of the reciprocals of these products is -7. Determine these three numbers.


Proposed by Efim I. Cherkassy, Rasba Institute, Israel, and Alexander A. Katz, St. John's University, NY. In a triangle ABC, AB is perpendicular to AC and the incircle touches BC at D. Prove that the area of ABC is equal to the product of BD and CD.


Proposed by U. I. Lydna, Beloretsk, Russia. In triangle ABC, BC^2 = AB(AB+AC). Prove that angle A is twice angle C without using trigonometry.


Proposed by Jerry Slocum, Beverly Hills. I had a set of seven tables arranged to form a square as in the standard Tangram puzzle depicted in the diagram on the left (see Math Horizons). One day, I found my tables rearranged to form a square with one-ninth of it missing, as depicted in the diagram on the right. At first, I thought someone had stolen the small square table, until I realized that the proportions were wrong. In fact, all seven tables were there. How were they arranged?

The Final Exam: Waving Goodbye