Constructions

Describe how to construct three circles which have areas in the ratio 1:2:3.

Explore patterns based on a rhombus. How can you enlarge the pattern - or explode it?

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.

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?

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

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

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

Without using a calculator, computer or tables find the exact values of cos36cos72 and also cos36 - cos72.

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?