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 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?
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?
Take three unit circles, each touching the other two. Construct
three circles $C_1$, $C_2$ and $C_3$, with radii $r_1$, $r_2$ and
$r_3$, respectively, as in the figure below. The circles that are
tangent to all three unit circles are $C_1$ and $C_3$, with $C_1$
the smaller of these. The circle through the three points of
tangency of the unit circles is $C_2$. Find the radii $r_1$, $r_2$
and $r_3$, and show that $r_1r_3=r_2^2$.