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Square Pair Circles
Age 16 to 18
Challenge Level 
For example
25 = 02 + ( ±5)2 = ( ±5)2 + 02 = ( ±3)2 + ( ±4)2 = ( ±4)2 + ( ±3)2
and this shows that there are 12 points with integer coordinates on the circle x2+ y2 = 52 with centre at the origin and radius 5 units.
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Baby Circle
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?
Orthogonal Circle
Given any three non intersecting circles in the plane find another circle or straight line which cuts all three circles orthogonally.
Ellipses
Here is a pattern for you to experiment with using graph drawing software. Find the equations of the graphs in the pattern.