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?
The angle between two circles at the point of intersection is
the angle between the tangents to the circles at that point
contained in the overlapping area or lune. Prove that, for any two
circles, the angles at both points of intersection are equal.
Three circles intersect at the point $D$ and, in pairs, at the
points $A,\ B$ and $C$ so that the arcs $AB$, $BC$ and $CA$ form a
curvilinear triangle with interior angles $\alpha$, $\beta$ and
$\gamma$ respectively. The diagrams show two possible cases. Prove
that, for any three such circles, $\alpha + \beta + \gamma =
When you have done this question you might like to consider what
happens on the surface of a sphere where instead of a flat surface
(Euclidean geometry) you are working on a surface with positive
curvature (Spherical or Elliptic geometry). See the article