Circle properties and circle theorems

problem
Favourite

Partly Circles

Age

14 to 16

Challenge level

What is the same and what is different about these circle questions? What connections can you make?
problem
Favourite

Baby Circle

Age

16 to 18

Challenge level

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?
problem
Favourite

Belt

Age

16 to 18

Challenge level

A belt of thin wire, length L, binds together two cylindrical welding rods, whose radii are R and r, by passing all the way around them both. Find L in terms of R and r.
problem

Incircles

Age

16 to 18

Challenge level

The incircles of 3, 4, 5 and of 5, 12, 13 right angled triangles have radii 1 and 2 units respectively. What about triangles with an inradius of 3, 4 or 5 or ...?
problem

Escriptions

Age

16 to 18

Challenge level

For any right-angled triangle find the radii of the three escribed circles touching the sides of the triangle externally.
problem

Not so Little X

Age

11 to 14

Challenge level

Two circles are enclosed by a rectangle 12 units by x units. The distance between the centres of the two circles is x/3 units. How big is x?
problem

Kissing

Age

16 to 18

Challenge level

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?
problem

Three Four Five

Age

14 to 16

Challenge level

Two semi-circles (each of radius 1/2) touch each other, and a semi-circle of radius 1 touches both of them. Find the radius of the circle which touches all three semi-circles.
problem

Tricircle

Age

14 to 16

Challenge level

The centre of the larger circle is at the midpoint of one side of an equilateral triangle and the circle touches the other two sides of the triangle. A smaller circle touches the larger circle and two sides of the triangle. If the small circle has radius 1 unit find the radius of the larger circle.
problem

Lunar Angles

Age

16 to 18

Challenge level

What is the sum of the angles of a triangle whose sides are circular arcs on a flat surface? What if the triangle is on the surface of a sphere?