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

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?

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?

Show that for any triangle it is always possible to construct 3 touching circles with centres at the vertices. Is it possible to construct touching circles centred at the vertices of any polygon?

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

Prove that a triangle with sides of length 5, 5 and 6 has the same area as a triangle with sides of length 5, 5 and 8. Find other pairs of non-congruent isosceles triangles which have equal areas.

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.

Medieval stonemasons used a method to construct octagons using ruler and compasses... Is the octagon regular? Proof please.

Move the ends of the lines at points B and D around the circle and find the relationship between the length of the line segments PA, PB, PC, and PD. The length of each of the line segments is. . . .

In the middle ages stone masons used a ruler and compasses method to construct exact octagons in a given square window. Open your compasses to a radius of half the diagonal of the square. . . .

If I print this page which shape will require the more yellow ink?

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

Triangle ABC has equilateral triangles drawn on its edges. Points P, Q and R are the centres of the equilateral triangles. What can you prove about the triangle PQR?

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 ?

We use statistics to give ourselves an informed view on a subject of interest. This problem explores how to scale countries on a map to represent characteristics other than land area.

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?

In the diagram the radius length is 10 units, OP is 8 units and OQ is 6 units. If the distance PQ is 5 units what is the distance P'Q' ?

Two perpendicular lines lie across each other and the end points are joined to form a quadrilateral. Eight ratios are defined, three are given but five need to be found.

A design is repeated endlessly along a line - rather like a stream of paper coming off a roll. Make a strip that matches itself after rotation, or after reflection

When a strip has vertical symmetry there always seems to be a second place where a mirror line could go. Perhaps you can find a design that has only one mirror line across it. Or, if you thought that. . . .

I noticed this about streamers that have rotation symmetry : if there was one centre of rotation there always seems to be a second centre that also worked. Can you find a design that has only. . . .