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.
Can you find the area of the central part of this shape? Can you do it in more than one way?
Show that there are infinitely many rational points on the unit circle and no rational points on the circle x^2+y^2=3.
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.