Circle properties and circle theorems

problem

Some(?) of the Parts

Age

14 to 16

Challenge level

2 out of 3

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
problem

Long Short

Age

14 to 16

Challenge level

2 out of 3

What can you say about the lengths of the sides of a quadrilateral whose vertices are on a unit circle?
problem

Three Balls

Age

14 to 16

Challenge level

2 out of 3

Do points P and Q lie inside, on, or outside this circle?
problem

Circle Box

Age

14 to 16

Challenge level

2 out of 3

It is obvious that we can fit four circles of diameter 1 unit in a square of side 2 without overlapping. What is the smallest square into which we can fit 3 circles of diameter 1 unit?
problem

Tied Up

Age

14 to 16

Challenge level

2 out of 3

How much of the field can the animals graze?
problem

Triangle Incircle Iteration

Age

14 to 16

Challenge level

3 out of 3

Keep constructing triangles in the incircle of the previous triangle. What happens?
problem

Encircling

Age

14 to 16

Challenge level

3 out of 3

An equilateral triangle is sitting on top of a square. What is the radius of the circle that circumscribes this shape?
problem

Cyclic Quad Jigsaw

Age

14 to 16

Challenge level

3 out of 3

A picture is made by joining five small quadrilaterals together to make a large quadrilateral. Is it possible to draw a similar picture if all the small quadrilaterals are cyclic?
problem

Similarly So

Age

14 to 16

Challenge level

3 out of 3

ABCD is a square. P is the midpoint of AB and is joined to C. A line from D perpendicular to PC meets the line at the point Q. Prove AQ = AD.
problem

Crescents and Triangles

Age

14 to 16

Challenge level

3 out of 3

Can you find a relationship between the area of the crescents and the area of the triangle?