The points P, Q, R and S are the midpoints of the edges of a non-convex quadrilateral.What do you notice about the quadrilateral PQRS and its area?

Join the midpoints of a quadrilateral to get a new quadrilateral. What is special about it?

Can you find the area of a parallelogram defined by two vectors?

Four rods, two of length a and two of length b, are linked to form a kite. The linkage is moveable so that the angles change. What is the maximum area of the kite?

Take an equilateral triangle and cut it into smaller pieces. What can you do with them?

The large rectangle is divided into a series of smaller quadrilaterals and triangles. Can you untangle what fractional part is represented by each of the ten numbered shapes?

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

How many questions do you need to identify my quadrilateral?

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?

Start with a triangle. Can you cut it up to make a rectangle?

Four rods are hinged at their ends to form a convex quadrilateral. Investigate the different shapes that the quadrilateral can take. Be patient this problem may be slow to load.

The diagonals of a trapezium divide it into four parts. Can you create a trapezium where three of those parts are equal in area?

We started drawing some quadrilaterals - can you complete them?

Can you recreate squares and rhombuses if you are only given a side or a diagonal?

A farmer has a field which is the shape of a trapezium as illustrated below. To increase his profits he wishes to grow two different crops. To do this he would like to divide the field into two. . . .

A game for 2 or more people, based on the traditional card game Rummy. Players aim to make two `tricks', where each trick has to consist of a picture of a shape, a name that describes that shape, and. . . .

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

A game in which players take it in turns to turn up two cards. If they can draw a triangle which satisfies both properties they win the pair of cards. And a few challenging questions to follow...

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

A game in which players take it in turns to try to draw quadrilaterals (or triangles) with particular properties. Is it possible to fill the game grid?

How many differently shaped rectangles can you build using these equilateral and isosceles triangles? Can you make a square?

A kite shaped lawn consists of an equilateral triangle ABC of side 130 feet and an isosceles triangle BCD in which BD and CD are of length 169 feet. A gardener has a motor mower which cuts strips of. . . .

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.