Work from the “Napoleon Triangle” series. Collage by Esther Ferrer in the late 1980s.Esther Ferrer
As we saw last week, complex numbers not only represent the discovery of buried treasure, but also represent a powerful mathematical tool despite their uncertain ontological status.
Using an approach similar to finding buried treasure, numerous geometric theorems can be proven, such as the famous “Napoleon Theorem.” The quotation marks indicate that the name should not be taken literally, since it is very doubtful that the author of the theorem was really Napoleon Bonaparte. Coxeter and Greitzer state in their book Geometry Revisited: “The possibility that Napoleon knew enough geometry to achieve this result is as questionable as the question of whether he knew enough English to write the famous palindrome ABLE WAS I ERE I SAW ELBA before I saw Elba). It is more likely that the theorem was proved by his friend Lorenzo Mascheroni or by another of the famous mathematicians with whom Bonaparte used to work, such as Laplace, Lagrange or Fourier. And some believe that this could have been proven a hundred years earlier by Torricelli or Fermat, who studied very similar geometric constructions. In any case, it has gone down in history as Napoleon’s theorem and goes like this:
If we construct two outer (or inner) equilateral triangles on the three sides of a triangle, the centers of these triangles will in turn be the vertices of an equilateral triangle (called Napoleon’s triangle).
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If you place the problem on the complex level, as we did with the treasure map, it is easy to prove the theorem; But it can also be attacked with other tools such as analytic geometry, trigonometry or from certain symmetries. I invite my clever readers to prove Napoleon’s theorem using their favorite tool.
And after proving it – or taking it for granted – it is not difficult to prove that the center of Napoleon’s triangle is the same as the centroid of the original triangle (remember that the centroid, centroid, or centroid of a triangle is the point of intersection ). its medians).
Napoleon’s problem
We must not confuse Napoleon’s theorem with Napoleon’s problem, proposed by him and solved by Mascheroni, which consists in dividing a circle into four equal parts (or, what is the same thing, finding the vertices of the inscribed square) using only one compass (aren’t you? dare to try it?). Mascheroni included it in his book Geometria del Compasso (1797), in which he showed that any geometric construction that can be done with a ruler and compass can also be done with just a compass. Mascheroni dedicated his influential book to his friend and protector Napoleon Bonaparte.
There are also some chess issues related to Napoleon, who was a big fan of the game and a more than acceptable chess player. One of the most famous is an artistic finale composed by Alexander Petroff in the 19th century. The author called it “Napoleon’s Retreat” because it was inspired by the defeat of the French army in the Battle of Moscow in 1812 (if you find the solution, you will understand why).
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