Investigate the properties of quadrilaterals which can be drawn with a circle just touching each side and another circle just touching each vertex.

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.

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?

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.

Can you find triangles on a 9-point circle? Can you work out their angles?

What is the relationship between the angle at the centre and the angles at the circumference, for angles which stand on the same arc? Can you prove it?

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?

A circle has centre O and angle POR = angle QOR. Construct tangents at P and Q meeting at T. Draw a circle with diameter OT. Do P and Q lie inside, or on, or outside this circle?

Explore when it is possible to construct a circle which just touches all four sides of a quadrilateral.

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

A circle of radius r touches two sides of a right angled triangle, sides x and y, and has its centre on the hypotenuse. Can you prove the formula linking x, y and r?

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

Find the missing angle between the two secants to the circle when the two angles at the centre subtended by the arcs created by the intersections of the secants and the circle are 50 and 120 degrees.

In the diagram the point P' can move to different places along the dotted line. Each position P' takes will fix a corresponding position for P. If P' moves along a straight line what does P do ?

Can you make a right-angled triangle on this peg-board by joining up three points round the edge?

This gives a short summary of the properties and theorems of cyclic quadrilaterals and links to some practical examples to be found elsewhere on the site.

In the diagram the point P can move to different places around the dotted circle. Each position P takes will fix a corresponding position for P'. As P moves around on that circle what will P' do?

M is any point on the line AB. Squares of side length AM and MB are constructed and their circumcircles intersect at P (and M). Prove that the lines AD and BE produced pass through P.

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

This set of resources for teachers offers interactive environments which support work on properties of angles in circles at Key Stage 4.

Three circular medallions fit in a rectangular box. Can you find the radius of the largest one?

The sides of a triangle are 25, 39 and 40 units of length. Find the diameter of the circumscribed circle.

This shape comprises four semi-circles. What is the relationship between the area of the shaded region and the area of the circle on AB as diameter?

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.

Draw some quadrilaterals on a 9-point circle and work out the angles. Is there a theorem?

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

Describe how to construct three circles which have areas in the ratio 1:2:3.

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. . . .

Which has the greatest area, a circle or a square inscribed in an isosceles, right angle triangle?

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

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

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?

Jennifer Piggott and Charlie Gilderdale describe a free interactive circular geoboard environment that can lead learners to pose mathematical questions.

Triangle ABC is equilateral. D, the midpoint of BC, is the centre of the semi-circle whose radius is R which touches AB and AC, as well as a smaller circle with radius r which also touches AB and AC. . . .

Use trigonometry to determine whether solar eclipses on earth can be perfect.