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?

Pericut

Two semicircle sit on the diameter of a semicircle centre O of twice their radius. Lines through O divide the perimeter into two parts. What can you say about the lengths of these two parts?

Kissing

Two perpendicular lines are tangential to two identical circles that touch. What is the largest circle that can be placed in between the two lines and the two circles and how would you construct it?

Ford Circles

Stage: 5 Challenge Level:

The proof of this striking result is elementary and uses a similar method to the problem Baby Circle.

 Given the radii of the circles and the points of contact with the x-axis you can always find the distances DC in terms of the radii. The circles will be tangent if and only if the distance AB is equal to the sum of the radii. The circles will be separate if and only if the distance AB is greater than the sum of the radii. It is possible to prove that the circles given can never overlap, that is the distance AB is never less than the sum of the radii.