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Napoleon’s Triangle

by
Nancy Perzel

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Napoleon’s Theorem states that given any triangle ABC, you can construct an equilateral triangle on each side of the triangle ABC so that the figure formed by connecting the centers of the three equilateral triangles will form an equilateral triangle. The equilateral triangle formed by these three centers is Napoleon's Triangle.

Napoleon Bonaparte is best known as a French military leader and emperor who conquered much of Europe in the early 19^th century. As a leader, Napoleon believed in the importance of science and education, and excelled in mathematics as a student. It is believed that Napoleon would have been equipped with enough knowledge of geometry to be able to prove the trigonometric theorem that bears his name; however, it’s unclear if he was the first to discover the result. Napoleon might never have discovered the steps in the proof; it could have simply been named in his honor. Either way, “the wonder doesn’t lay in who first proved it, but in the uniquely human ingenuity required to discover, and appreciate, its beauty.”

http://www.uh.edu/engines/epi2550.htm

http://www.history.com/topics/napoleon

Now we will look at Napoleon's triangle and prove that it is indeed an equilateral triangle.

Let’s begin with a triangle ABC.

Now we will construct three equilateral triangles on each side of triangle ABC.

Next we will find the center, or centroid, of each of the three equilateral triangles.

Step 1: Construct a midpoint for each side of the triangles.

Step 2: Construct a segment connecting the midpoint of a side of the equilateral triangle to the opposite vertex. Do this for each midpoint, creating three segments. The point of intersection is the center of the equilateral triangle.

Find the center of all three equilateral triangles.

Finally, we will connect these three points to form Napoleon’s Triangle.

The yellow triangle is known as Napoleon’s Triangle.

From simply looking at the construction it appears that Napoleon’s Triangle is indeed equilateral.  Geometer's Sketchpad is a great tool to use for measurements that will later reinforce a proof.

An equilateral triangle is defined as a triangle with three equal sides. An equilateral triangle is also equiangular, since its interior angles all equal 60°. All equilateral triangles are also equiangular.

Dictionary - http://intermath.coe.uga.edu/n

The interior angles of Napoleon’s Triangle, triangle DEF, have equal measures, m<D = m<E = m<F. Each angle measures 60°. Additionally, the lengths of each side of the triangle are equal, DE = FD = EF. Therefore, Napoleon’s Triangle is in fact an equilateral triangle.

Link to GSP sketch of Napoleon’s Triangle with animation, click here.

For more advanced proofs of Napoleon’s Theorem, visit the following webpage: http://www.cut-the-knot.org/proofs/napoleon.shtml

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