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?
Given any three non intersecting circles in the plane find another
circle or straight line which cuts all three circles orthogonally.
Here is a pattern for you to experiment with using graph drawing
software. Find the equations of the graphs in the pattern.
All powers of 5 can be written as the sum of two square numbers
in different ways. Investigate the number of points with integer
coordinates on circles with centres at the origin for which the
square of the radius is a power of 5. For the two circles shown in
red in the diagram the radius is a power of 5 and for the circles
shown in black this is not so but the square of the radius is a
power of 5.