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Try to visualise necklaces of touching circles centred at the
vertices of a rightangled 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 rightangled 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 rightangled triangle $ABC$ find the radii of the
three escribed circles as shown in the diagram.

A small circle fits between two touching circles so that all three circles touch each other and have a common tangent? What is the exact radius of the smallest circle?
Two perpendicular lines are tangential to two identical circles that touch. What is the largest circle that can be placed in between the two lines and the two circles and how would you construct it?
Ten squares form regular rings either with adjacent or opposite vertices touching. Calculate the inner and outer radii of the rings that surround the squares.