Prove that, given any three parallel lines, an equilateral triangle always exists with one vertex on each of the three lines.

The diagonal of a square intersects the line joining one of the unused corners to the midpoint of the opposite side. What do you notice about the line segments produced?

What fractions can you divide the diagonal of a square into by simple folding?

Drawing a triangle is not always as easy as you might think!

Have a go at creating these images based on circles. What do you notice about the areas of the different sections?

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

How can you represent the curvature of a cylinder on a flat piece of paper?

What shape and size of drinks mat is best for flipping and catching?

Design and construct a prototype intercooler which will satisfy agreed quality control constraints.

Investigate constructible images which contain rational areas.

Using a ruler, pencil and compasses only, it is possible to construct a square inside any triangle so that all four vertices touch the sides of the triangle.

Jenny Murray describes the mathematical processes behind making patchwork in this article for students.

You are only given the three midpoints of the sides of a triangle. How can you construct the original triangle?

You are given a circle with centre O. Describe how to construct with a straight edge and a pair of compasses, two other circles centre O so that the three circles have areas in the ratio 1:2:3.

The challenge is to produce elegant solutions. Elegance here implies simplicity. The focus is on rhombi, in particular those formed by jointing two equilateral triangles along an edge.

Draw a line (considered endless in both directions), put a point somewhere on each side of the line. Label these points A and B. Use a geometric construction to locate a point, P, on the line,. . . .

Construct a line parallel to one side of a triangle so that the triangle is divided into two equal areas.