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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.

Curved Square

Age 16 to 18 Challenge Level:
You can solve this using basic properties of areas of circles, coordinate geometry and Pythagoras's theorem or simple trigonometry.
This worksheet suggests a possible method.

You can also solve it using calculus.
This worksheet suggests a possible method.