Two perpendicular lines are tangential to two identical circles that touch. What is the largest circle that can be placed in between the two lines and the two circles and how would you construct it?

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,. . . .

Draw a square and an arc of a circle and construct the Golden rectangle. Find the value of the Golden Ratio.

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

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.

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

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

Explain how to construct a regular pentagon accurately using a straight edge and compass.

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.

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.

Investigate constructible images which contain rational areas.

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

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

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?

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