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?
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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

Fixing It

Age

16 to 18

Challenge level

A and B are two fixed points on a circle and RS is a variable diamater. What is the locus of the intersection P of AR and BS?
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

Coins on a Plate

Age

11 to 14

Challenge level

Points A, B and C are the centres of three circles, each one of which touches the other two. Prove that the perimeter of the triangle ABC is equal to the diameter of the largest circle.
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

Circles in Circles

Age

16 to 18

Challenge level

This pattern of six circles contains three unit circles. Work out the radii of the other three circles and the relationship between them.
problem

Some(?) of the Parts

Age

14 to 16

Challenge level

A circle touches the lines OA, OB and AB where OA and OB are perpendicular. Show that the diameter of the circle is equal to the perimeter of the triangle