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The idea for this problem was suggested by Geoff Faux. See also the NRICH Problem 'Polycircles' .
Escriptions
Age 16 to 18
Challenge Level 
The idea for this problem was suggested by Geoff Faux. See also the NRICH Problem 'Polycircles' .
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Try to visualise necklaces of touching circles centred at the
vertices of a right-angled triangle, first where the circles touch
on the sides of the triangle and then where points of contact lie
on the sides produced. Now try to visualise where these circles
meet the inscribed and escribed circles of the triangle.
Given any right-angled triangle $ABC$ with sides $a, b$ and
$c$, find the radii of the three circles with centres at $A, B$ and
$C$ such that each circle touches the other two and two of the
circles touch on $AB$ between $A$ and $B$, two circles touch on
$CA$ produced and two circles touch on $CB$ produced.
Now for any right-angled triangle $ABC$ find the radii of the
three escribed circles as shown in the diagram.
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