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?

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?

Ten squares form regular rings either with adjacent or opposite
vertices touching. Calculate the inner and outer radii of the rings
that surround the squares.

Retracircles

Age 16 to 18 Challenge Level

The circles centres $O$, $A$, $B$ and $C$ touch one another.
Prove that the radii of the circles with centres $A$, $B$ and $C$
are in the ratio $1 : 2 : 3$ and that the triangles $OBC$ and $ABC$
are right angled $3 : 4 : 5$ triangles.