Math 487 Lab 10 (Napoleon Theorem)

The Napoleon Figure

Construct the following figure. It may go faster if you create a tool 'Eqtri with center from side' for constructing a equilateral triangle ABC with center O given points A and B (i.e., construct the triangle from the edge, but also construct the center and hide the construction lines).

Use this tool to make a Napoleon figure like this one.

The Napoleon Theorem by Experiment

• Construct the circle XYZ. drag A, B and C around and see what kind of triangle this is.
• Do some measuring to provide extra evidence. (This makes it pretty certain that you conjecture is correct, but it does not prove it or explain why yet.)
• Does the relationship still hold if you drag A across line BC so that the triangles are on the inside?
• What if A is on segment BC?

Napoleon's theorem states that XYZ is equilateral for any choice of ABC (

Why is this Napoleon's Theorem? It is claimed (probably not entirely accurately) that this theorem was discovered by Emperor Napoleon of France. What is really true is that he did have a taste for mathematics and also surrounded himself with quite a few mathematicians in his government. He also set up some French schools like the Polytechnique, that are still very important scientific institutions today. And it makes for a very memorable name for the theorem.

Proof based on transformations

Part 1.  Investigating the Products of 120-degree rotations

In the proof we will compose three 120-degree rotatoins. So on a new, separate page, let's review what this will look like.

• Draw 3 points X, Y, Z on the plane and also draw a point P.
• Rotate P with center X by 120 degrees to get P'.
• Rotate P' with center Y by 120 degrees to get P''. Thus P'' = Y₁₂₀X₁₂₀(P). What kind of isometry does our general theory say Y₁₂₀X₁₂₀ is? Can you see this? Can you see the angle and the center?
• Rotate P'' with center Z by 120 degrees to get P'''. Thus P'' = Z₁₂₀Y₁₂₀X₁₂₀(P). What kind of isometry does our general theory say Z₁₂₀Y₁₂₀X₁₂₀ is? Can you see this?
• To visualize the relationship between P and P''', construct the segment PP'''.  Leaving X, Y and Z fixed, drag P around and see how the segment PP''' changes.  Does the length change?  Does the direction change?

Question:  What kind of isometry is the transformation that takes P to P'''.  This isometry is the composition T = Z₁₂₀Y₁₂₀X₁₂₀.

Answer ___________________________

• Move one of X, Y or Z and again observe how PP''' changes as P moves.  Is the transformation still the same kind?  Is it exactly the same transformation? 
• Now our goal is to make P''' coincide with P.  It is does it for one P, then it should do it for any position of P.  Leaving X and Y fixed, drag Z until P''' coincides with P. 

Question: Where is Z when this occurs?

Answer ____________________

• How is this consistent with what we have learned about computing Y₁₂₀X₁₂₀?

Part 2.  The proof of Napoleon's Theorem

• Let P = B in the figure, then P' = A.

What is P'' in this figure if P' = A? _________________

What is P''' in this figure given the P'' above? _________________

• Since in this figure P = P''', based on your conclusions from the last part, what can you say about the triangle XYZ?

Answer: Triangle XYZ is __________________________________

• Draw triangle XYZ in your figure and see that the conclusion appears to be correct as you drag A, B, and C around and XYZ moves in consequence.

Part 3.  Building a Napoleon Figure with Rotations

In a new sketch, draw a point X and a point Y.  Let Z be the rotation of X with center Y by 60 degrees.

• Then draw a point P in the sketch.  Repeat the construction of P', P'', and P''' from Part 1.  You should find that P''' = P, so you can construct segments PP', P'P'', P''P''' to form a triangle. (If this does not happen, ask for help!)  Call the points P = B, P' = A, P'' = C so your figure matches the figure on the first page.
• Now construct the 3 equilateral triangles in the figure by rotating appropriate points by 120 degrees with centers X, Y, Z.  Construct the equilateral triangle interiors and color the 3 triangles in 3 different colors.
• You can now drag point B and see how the figure changes with X, Y, and Z immobile.

Part 4.  Covering the plane by a Napoleon tessellation

• Continuing with this sketch from Part 3, begin a tessellation of the plane by rotating the whole figure by 120 degrees, first with center X.
• Then select the whole figure and rotate by 120 with center Y.
• Then select the whole figure and rotate by 120 with center Z.
• Then select the whole figure and rotate by 120 with center X.
• Continue until the pattern becomes clear.

Question:  What are the symmetries of this tessellation?  Does this suggest another reason why the points X, Y, Z have to form an equilateral triangle?