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?

Retracircles

Stage: 5 Challenge Level:

$BO$ is a tangent to the two equal circles and hence angle $BOC$ is a right angle. If $OC=$3 units then the large outer circle has radius $6$ units. Pythagoras' theorem will give the radius of the circle centre $B$. You cannot assume that angle $BAC$ is a right angle.