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Bang's Theorem

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Rudolff's Problem

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Polycircles

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Why do this problem?
This problem is accessible for different age groups. It is a rich problem that provides suitable challenges for a wide age range. Younger learners can find solutions for three circles given the sides of a triangles by trial and improvement.

Once the learners can solve two simultaneous linear equations in two unknowns then they can be challenged to solve this problem by solving three simultaneous equations. Because the coefficients are all unity the equations are easy to solve.

This problem is often used at stages 3 as well as 4. At Stage 5 it is useful as an application of solving simultaneous linear equations and the question about which polygons have solutions and which do not can take learners into linear algebra.

The problem demonstrates a powerful inter-relationship between geometry and algebra. It also links to the problem Pentagon from 1926 which has a different context but where the mathematics is identical giving an experience of isomorphism in mathematics.

Possible approaches
For younger learners who know how to solve a pair of simultaneous linear equations this problem provides a good series of challenges. Taking a numerical example where the lengths of the sides of the triangle are known (see the Hint), and the radii of the polycircles has to be found, then three simultaneous linear equations can easily be found and solved. Even though learners may only have been taught to solve two simultaneous equations in two unknowns many will be able to solve these three equations for themselves and get satisfaction from being able to do so independently. The results can be checked by drawing.

The next step is to generalise from a particular numerical example and to use exactly the same steps in the algebra to derive formulae for the radii in terms of the lengths of the sides of the triangle.This is a good exercise in a concrete setting for working with algebra.

Learners who have succeeded so far can explore the existence of polycircles for quadrilaterals and pentagons and how the algebra explains the different geometrical phenomena that arise.

For older learners who have been been introduced to linear algebra, the generalisation of the problem to polygons with n sides provides a challenge to explain why there are unique solutions for certain values of n and not for others.

Possible support
The problem is written to start with a numerical special case of a triangle in order to support younger and less confident learners.They may first try trial and improvement where the problem is like arithmagons. From that they may be able to develop a method for finding solutions in the general case and even for getting the formula without the use of simultaneous equations.

Possible extension
You could pose the problem for general polygons and leave the learners to decide for themselves whether or not to start with special cases.

The problem Pentagon from 1926 is a natural extension of this problem.