The Math Forum || Annie's Sketchpad Activites || Java Version

# Napoleon's Theorem

This is an investigation of a theorem attributed to Napoleon Bonaparte (yes, the same person we associate with Waterloo).

1. Construct a triangle, any triangle. You can do this using the segment tool. Then, using the selection arrow, drag each vertex of the triangle to make sure that everything is connected the way you want it to be. Now select the three segments and change their weight to thick and their color to blue (both options are under the Display menu).

2. Construct equilateral triangles on the sides of the triangles. You can do this by constructing a circle centered at one vertex that goes through the other vertex, then do another circle the other way. You can hide the circles by selecting them with the selection arrow and choosing Hide under the Display menu. (You might also have a script tool that constructs the triangle.)
3. Find the centers of these equilateral triangles. You can find the center of a triangle by connecting two of the vertices to the midpoint of the opposite side. Midpoint is an option under the Construct menu. Connect the centers with segments. Color these three new segments red. What is true of them? (You can hide the midpoints and segments you drew to find the centers by using Hide under the Display menu.)

4. Connect the outside vertices of the equilateral triangles with the opposite vertex of the original triangle (only one is shown in the picture). Color these line segments green - you might make them thick as well. What is true of the line segments? (See if you can find at least two things.)

5. (tough question - you might try 6 first) These three segments intersect at a point. How is this point related to the original triangle?

6. Reflect each vertex of the Napoleonic triangle (the red one) over the closest edge of the original triangle (the blue one). To do this, select an edge of the original triangle and choose Mark Mirror from the Transform menu (you can also double-click on the segment to mark it as a mirror). Now select the closest vertex of the red triangle and choose Reflect from the Transform menu. Do this for the other two vertices and connect the three new points to form a triangle. Color the segments pink. What is true of this triangle?

7. How do the areas of the pink triangle and the red triangle compare to the area of the blue triangle?