Solutions for Mathematics Enrichment

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## The Arbelos logo

Our logo is a reference to a problem solved by Archimedes: find the radius of the inscribed circle of an arbelos in terms of the radii of the three semicircles forming the figure. Further details of this problem and other properties of the arbelos appear below. In the logo, the inscribed circle and arbelos have radii in the ratio 6:7:14:21.

## The geometry of the arbelos

The region enclosed by three touching semicircles with diameters on the same line (the region shown shaded blue in the diagram) is known as an arbelos, from the ancient Greek word for a shoemaker's knife.

Archimedes gave the name to the figure and was one of the first to investigate its mathematical properties. In Liber Assumptorum, Proposition 4, he proved that the area of the arbelos is equal to the area of the circle (shown shaded red in the diagram) which is determined by either of the two diameters indicated.

Many other results about the arbelos were discovered in ancient times, especially by Archimedes and, later, Pappus. There has been a resurgence of interest in the arbelos in modern times, with many new results being discovered.

To learn more about properties of the arbelos, Thomas Schoch’s site Arbelos – Amazing Properties is recommended. He also includes a comprehensive list of references.

“… geometry is nothing at all, if not a branch of art …”—Julian Coolidge

Andrew Jobbings gained both his BSc and his PhD in mathematics from Durham University. He taught mathematics for 28 years, including 14 years as Head of Department at Bradford Grammar School, before founding Arbelos.

With a keen interest in providing mathematics enrichment activities, Andrew gives Royal Institution masterclasses and helps at UKMT Mathematical Circles and Summer Schools. He devises problems for the UKMT Maths Challenges and other competitions, and designs the key fob problems. He has regularly chaired a problems group for the European Kangaroo and also helped to establish the IMOK Olympiad for the UKMT.

In addition to his work for Arbelos, he also presents sessions at teacher meetings, acts as a consultant for e-learning and other educational projects, and undertakes commissions.

The Beauty of Mathematics poster collection developed from posters he created for use in his own classroom. For ClubMaths, he researched the mathematics behind the activities and devised the problems.

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##### The incircle

The inscribed circle of an arbelos is tangent to all three semicircles.

In the diagram on the left, the incircle is shown with light blue circumference within the blue arbelos.

Archimedes found an expression for the radius of the incircle. In modern notation the formula for the radius is $\frac{ab(a+b)}{a^2+ab+b^2},$ where $$a$$ and $$b$$ are the radii of the two smaller semicircles.

“These interesting designs use great graphics skills and colours which are pleasing to the eye.

My favourite design is the Lemniscate of Bernoulli … because it looks complex, making a very striking pattern.”

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