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

#### S-80.

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.

#### S-81.

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.

#### S-82.

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.

#### S-83.

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?