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##### Stage: 4 Challenge Level:

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

##### Stage: 3 and 4

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.

##### Stage: 3 Challenge Level:

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?

### Bi-cyclics

##### Stage: 4 Challenge Level:

Two circles intersect at A and B. Points C and D move round one circle. CA and DB cut the other circle at E and F. What do you notice about the line segments CD and EF?

### Long Short

##### Stage: 4 Challenge Level:

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

##### Stage: 4 Challenge Level:

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.

##### Stage: 4 Challenge Level:

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

##### Stage: 4 Challenge Level:

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?

### Diagonals for Area

##### Stage: 4 Challenge Level:

Can you prove this formula for finding the area of a quadrilateral from its diagonals?

##### Stage: 3 Challenge Level:

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. . . .

### An Equilateral Triangular Problem

##### Stage: 3 Challenge Level:

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

### Fractions Rectangle

##### Stage: 3 Challenge Level:

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?

### Shapely Pairs

##### Stage: 3 Challenge Level:

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...

### Property Chart

##### Stage: 3 Challenge Level:

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?

##### Stage: 3 Challenge Level:

We started drawing some quadrilaterals - can you complete them?

##### Stage: 3 Challenge Level:

How many questions do you need to identify my quadrilateral?

### Making Rectangles, Making Squares

##### Stage: 3 Challenge Level:

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

### Kite

##### Stage: 3 Challenge Level:

Derive a formula for finding the area of any kite.