Given any triangle with sides a, b and c, is it always
possible to construct 3 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?
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For any regular polygon of
side length a units, it
is always possible to draw
circles with centres at the
vertices of the polygon and
radii
units
to form a "polycircle" in
which each circle just
touches its neighbours.
Investigate and explain what
happens in the case of
non-regular polygons.
Experiment with the
interactivities below or do
the construction for yourself
with ruler and compasses or
using dyanamic geometry
software.
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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?
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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.
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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.