Napoleon's theorem

Your text book(s) contain instructions on how to perform constructions such as bisecting angles, drawing perpendiculars etc.
However the exercises they give are normally fairly dry. These pages show you some constructions you can apply those skills to and are hopefully a little more engaging.

This construction illustrates Napoleon's theorem.

Amongst other interests (such as world domination...) Napoleon was a fan of mathematics. He is even credited with the following theorem:

If you construct equilateral triangles on the edges of any triangle then their centres will form the vertices of another equilateral triangle.

Below you will find instructions on how you can verify the theorem with a triangle of your own. There is also an interactive example to help guide you through the construction.

  1. Construct an equilateral triangle on each side of the triangle.
  2. Bisect the angles of the equilateral triangles to find the centres.
  3. Join the centres to form a triangle. Check it is equilateral by measuring the sides or the angles.

Note: the constructions are only shown for one triangle to keep the diagram simpler.

To download this Geogebra file click here.