Commandino's theorem, named after Federico Commandino (1509–1575), 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.

History 
The theorem is attributed to Commandino, who stated, in his work De Centro Gravitatis Solidorum (The Center of Gravity of Solids, 1565), that the four medians of the tetrahedron are concurrent. However, according to the 19th century scholar Guillaume Libri, Francesco Maurolico (1494–1575) 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:

 Let  be a -simplex of some dimension  in  and let  be its vertices. Furthermore, let , be the medians of , the lines joining each vertex  with the centroid of the opposite -dimensional facet . Then, these lines intersect each other in a point , in a ratio of .

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:

 Let  and  be natural numbers, so that in an -vector space ,  pairwise different points  are given.
 Let  be the centroid of the points , let  be the centroid of the points , and let  be the centroid of all of these  points.
 Then, one has

 In particular, the centroid  lies on the line  and divides it in a ratio of .

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 :

 One may find the centroid of a tetrahedron by taking the midpoints of two pairs of two of its opposite edges and connecting the corresponding midpoints through their respective midline. The intersection point of both midlines will be the centroid of the tetrahedron.

Since a tetrahedron has six edges in three opposite pairs, one obtains the following corollary:

In a tetrahedron, the three midlines corresponding to opposite edge midpoints are concurrent, and their intersection point is the centroid of the tetrahedron.

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:

Let a quadrilateral in  be given. Then the two midlines connecting opposite edge midpoints intersect in the centroid of the quadrilateral and are divided in half by it.

References

External links 

 
 A Couple of Nice Extensions of the Median Properties

Theorems in geometry
Euclidean geometry