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Bang's Theorem

If all the faces of a tetrahedron have the same perimeter then show that they are all congruent.

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Rudolff's Problem

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

tri-circle quad-circle pent-circle
For any regular polygon it is always possible to draw circles with centres at the vertices of the polygon, and radii equal to half the length of the edges, to form a "polycircle" in which each circle just touches its neighbours.

Investigate and explain what happens in the case of non-regular polygons. Is it always possible to construct three circles with centres at the vertices of the triangle so that the circles just touch?

Try a numerical example, say the 3 points give a triangle of sides 12, 15 and 13. Find the radii of the circles.

Now use the same method when the sides of the triangle are $a$, $b$ and $c$.

Generalise still more! What about 4 circles? 5 circles? $n$ circles?

Experiment with the interactivities below or do the construction for yourself with ruler and compasses or using dyanamic geometry software.

The next applet is a 5-sided polygon. Again you can check that wherever you move the vertices off the polygon, the circles centred at those points will remain tangent to their two adjacant circles. Can you calculate the radii given the lengths of the sides of the polygon?

In the final applet below we have a quadrilateral. Here you see that there are 4-sided polygons where the 4 circles are not all tangent to one another. Can you make the red and green circles lie on top of one another? Then you have made all the circles tangent to their neighbours.

Created with GeoGebra

For the final challenge, what is the locus of the centre of the red and green circles when it moves so that these circles are coincident, always touching their neighbours, maintaining a perfect polycircle?

To investigate this problem further, download a copy of GeoGebra and experiment for yourself. Geogebra is free educational mathematics software that is very easy to use and combines dynamic geometry, coordinate geometry, algebra and calculus. You can also download the Quickstart guide for beginners.