Given any three non intersecting circles in the plane find another
circle or straight line which cuts all three circles orthogonally.
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?
Can you find the area of the central part of this shape? Can you do it in more than one way?
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.
Show that there are infinitely many rational points on the unit
circle and no rational points on the circle x^2+y^2=3.
Here is a pattern for you to experiment with using graph drawing
software. Find the equations of the graphs in the pattern.