Commandino's theorem

Commandino's theorem has a direct analog for simplexes of any dimension:^[6]

Let ${\displaystyle \Delta }$ be a ${\displaystyle d}$-simplex of some dimension ${\displaystyle d>1}$ in ${\displaystyle \mathbb {R} ^{n}\;(d,n\in \mathbb {N} ,n\geq d)}$ and let ${\displaystyle V_{0},V_{1},\ldots ,V_{p}}$ be its vertices. Furthermore, let ${\displaystyle \ell _{0},\ell _{1},\ldots ,\ell _{d}}$, be the medians of ${\displaystyle \Delta }$, the lines joining each vertex ${\displaystyle V_{i}}$ with the centroid of the opposite ${\displaystyle (d-1)}$-dimensional facet ${\displaystyle V_{0}\ldots V_{i-1}V_{i+1}\ldots V_{d}}$. Then, these lines intersect each other in a point ${\displaystyle S}$, in a ratio of ${\displaystyle d:1}$.

The former analog is easy to prove via the following, more general result, which is analogous to the way levers in physics work:^[7]

Let ${\displaystyle m}$ and ${\displaystyle k}$ be natural numbers, so that in an ${\displaystyle \mathbb {R} }$-vector space ${\displaystyle {\mathcal {V}}}$, ${\displaystyle m+k}$ pairwise different points ${\displaystyle X_{1},\dots ,X_{m},Y_{1},\dots ,Y_{k}\in {\mathcal {V}}}$ are given.

Let ${\displaystyle S_{X}}$ be the centroid of the points ${\displaystyle X_{i}\;(i=1,\dots ,m)}$, let ${\displaystyle S_{Y}}$ be the centroid of the points ${\displaystyle Y_{j}\;(j=1,\dots ,k)}$, and let ${\displaystyle S}$ be the centroid of all of these ${\displaystyle m+k}$ points.

Then, one has

${\displaystyle S=S_{X}+{\frac {k}{m+k}}(S_{Y}-S_{X})={\frac {m}{m+k}}S_{X}+{\frac {k}{m+k}}S_{Y}.}$

In particular, the centroid ${\displaystyle S}$ lies on the line ${\displaystyle {\overline {{S_{X}}{S_{Y}}}}}$ and divides it in a ratio of ${\displaystyle k:m}$.

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 [de]:^[8][9]

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:^[8]

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

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:^[10][11]

Let a quadrilateral in ${\displaystyle \mathbb {R} ^{2}}$ be given. Then the two midlines connecting opposite edge midpoints intersect in the centroid of the quadrilateral and are divided in half by it.

• Weisstein, Eric W. "Commandino's Theorem". MathWorld.
• A Couple of Nice Extensions of the Median Properties