and the Napoleon Points

Napoleon's Theorem is the name popularly given to a theorem which states that if equilateral triangles are constructed on the three legs of any triangle, the centers of the three new triangles will also form an equilateral triangle. In the figure the original triangle is labeled A, B, C, and the centers of the three equilateral triangles are A', B', C'. If the segments from A to A', B to B', and C to C' are drawn they always intersect in a single point, called the First Napoleon Point. If the three equilateral triangles are drawn interior to the original triangle, the centers will still form an equilateral triangle, but the segments connecting the centers with the opposite vertices of the original triangle meet in a (usually) different point, called the 2nd Napoleon Point.

Although it is known that Napoleon had a keen interest in geometry, math historians seem unable to find evidence he really discovered the theorem. Here is a letter on the subject from Antreas P. Hatzipolakis, a real living Greek mathematician, to the Geometry Forum.

John E. Wetzel published an article in the _AMMonthly_ (1992, 339ff) on the "Converses of Napoleon's Theorem" In the paragraph "Why Napoleon?" we read:

The early history of Napoleon's theorem and the Fermat points F, F' (which are also called isogonic centers of ABC) is summarized in Mackey [21], who traces the fact that LMN and L'M'N' are equilateral to 1825 to one Dr. W. Rutherford [27] and remarks that the result is probably older.

References: 21. J. S. Mackay, Isogonic centres of a triangle. Proc. Edinburgh Math. Society 15(1897) 110-118

27. W. Rutherford, VII. Quest. (1439), Ladies' Diary No. 122(1825) 47.

The theorem is a special case of a more general theorem that says if the three constructed triangles are similar to each other and in the same orientation, then the centers will form a triangle that is also similar to the newly constructed triangles. The existance of a concurrent point for the three segments joining the centers to the opposite vertices is also a general property of the construction.

Napoleon's Theorem can be seen as related to a theorem about quadrilaterals that is credited to H. van Aubel, described by Dick Klingens as "a very well-known triangle guru of the 19th century, who published many short papers on Euclidean geometry. I have lots of these papers, most of them published in the Mathesis Journal. Van Aubel taught elementary mathematics at the Athenee in Antwerp ( Belgium)." Later a note from ... in Malta, cites the first publicaton of Van Aubel's theorem thus, "It seems that it first appeared in Nouvelles Corresp. mathematique 4 (1878) pp 40-44."

A very nice illustration of the proof is presented by Antonio Gutierrez here.

In another note Dick Klingens observed on the relationship between the two theorems, and added another note that introduced me to two related theorems: "When you construct squares on the sides of a triangle (all outwardly), the three lines connecting the centers of the squares are concurrent in the (first) Vecten point. The three lines are the altitudes of the triangle with those centers as vertices. In this case one side of the "Van Aubel quadrilateral" is of zero length. When the squares are constructed all inwardly, the theorem holds. The point of concurrency is called the second Vecten point."

Vecten was a French mathematician who taught in Nimes. I have no further information on Vecten,
except that he published this theorem in 1817 in Gergonne's Journal. Again I lean on the work of Antonio for an illustraion. You can see the proof at his site.

For those who want to progress another step down the line, A generalization of van Aubel's theorem to Rhombii on the edges is in this October copy of Mathematics Magazine from the MAA. It is in PDF format.