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

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

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?

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

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

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

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

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.

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

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?

Investigate constructible images which contain rational areas.

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,. . . .

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

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.

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