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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?

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Orthogonal Circle

Given any three non intersecting circles in the plane find another circle or straight line which cuts all three circles orthogonally.

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Ellipses

Here is a pattern for you to experiment with using graph drawing software. Find the equations of the graphs in the pattern.

Square Pair Circles

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

Circles for which the square of the radii are 5, 25, 125, 625 and 3125.

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.