Commandino's theorem


Commandino's theorem, named after Federico Commandino, states that the four medians of a tetrahedron are concurrent at a point S, which divides them in a 3:1 ratio. In a tetrahedron a median is a line segment that connects a vertex with the centroid of the opposite face – that is, the centroid of the opposite triangle. The point S is also the centroid of the tetrahedron.
The theorem is attributed to Commandino, who stated, in his work De Centro Gravitatis Solidorum, that the four medians of the tetrahedron are concurrent. However, according to the 19th century scholar Guillaume Libri, Francesco Maurolico claimed to have found the result earlier. Libri nevertheless thought that it had been known even earlier to Leonardo da Vinci, who seemed to have used it in his work. Julian Coolidge shared that assessment but pointed out that he couldn't find any explicit description or mathematical treatment of the theorem in da Vinci's works. Other scholars have speculated that the result may have already been known to Greek mathematicians during antiquity.

Generalizations

Commandino's theorem has a direct analog for simplexes of any dimension:

Full generality

The former analog is easy to prove via the following, more general result, which is analogous to the way levers in physics work:

Reusch's theorem

The previous theorem has further interesting consequences other than the aforementioned generalization of Commandino's theorem. It can be used to prove the following theorem about the centroid of a tetrahedron, first described in the Mathematische Unterhaltungen by the German physicist Friedrich Eduard Reusch:
Since a tetrahedron has six edges in three opposite pairs, one obtains the following corollary:

Varignon's theorem

A specific case of Reusch's theorem where all four vertices of a tetrahedron are coplanar and lie on a single plane, thereby degenerating into a quadrilateral, Varignon's theorem, named after Pierre Varignon, states the following: