Commandino's theorem

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.Nathan Altshiller Court: Notes on the centroid. The Mathematics Teacher, Vol. 53, No. 1 (JANUARY 1960), pp. 34 ([https://www.jstor.org/stable/27956057 JSTOR]) Other scholars have speculated that the result may have already been known to Greek mathematicians during antiquity.Howard Eves: Great Moments in Mathematics (before 1650). MAA, 1983, {{ISBN|9780883853108}}, p. 225

Commandino's theorem has a direct analog for simplexes of any dimension:{{Cite book|author=Egbert Harzheim|title=Einführung in die kombinatorische Topologie|date=1978|publisher=Wissenschaftliche Buchgesellschaft|ISBN=3-534-07016-X|place=Darmstadt|pages=33|language=de}}

: Let $\backslash Delta$ be a $d$-simplex of some dimension $d>1$ in $\backslash R^n\; \backslash ;\; (d,n\; \backslash in\; \backslash N\; ,\; n\; \backslash geq\; d)$ and let $V\_0,V\_1,\backslash ldots,V\_p$ be its vertices. Furthermore, let $\backslash ell\_0,\; \backslash ell\_1,\backslash ldots,\backslash ell\_d$, be the medians of $\backslash Delta$, the lines joining each vertex $V\_i$ with the centroid of the opposite $(d-1)$-dimensional facet $V\_0\backslash ldots\; V\_\{i-1\}V\_\{i+1\}\backslash ldots\; V\_d$. Then, these lines intersect each other in a point $S$, in a ratio of $d:1$.

The former analog is easy to prove via the following, more general result, which is analogous to the way levers in physics work:{{citation|surname1=Egbert Harzheim|title=Einführung in die Kombinatorische Topologie|publication-place=Darmstadt|at=p. 31|isbn=3-534-07016-X|date=1978|language=de

}}

: Let $m$ and $k$ be natural numbers, so that in an $\backslash R$-vector space $\backslash mathcal\; \{V\}$, $m+k$ pairwise different points $X\_1,\; \backslash dots,\; X\_m,\; Y\_1,\; \backslash dots,\; Y\_k\; \backslash in\; \backslash mathcal\; \{V\}$ are given.

: Let $S\_X$ be the centroid of the points $X\_i\; \backslash ;\; (i=1,\; \backslash dots,\; m)$, let $S\_Y$ be the centroid of the points $Y\_j\; \backslash ;\; (j=1,\; \backslash dots,\; k)$, and let $S$ be the centroid of all of these $m+k$ points.

: Then, one has

::$S\; =\; S\_X\; +\; \backslash frac\{k\}\{m+k\}\; (S\_Y-S\_X)\; =\; \backslash frac\{m\}\{m+k\}\; S\_X\; +\; \backslash frac\{k\}\{m+k\}\; S\_Y.$

: In particular, the centroid $S$ lies on the line $\backslash overline\{\; \{S\_X\}\; \{S\_Y\}\}$ and divides it in a ratio of $k:m$.

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 {{ill|Friedrich Eduard Reusch|de|Friedrich Eduard Reusch}}:Friedrich Joseph Pythagoras Riecke (Hrsg.): Mathematische Unterhaltungen. Zweites Heft. 1973, S. 100, 128In den Mathematische Unterhaltungen (Zweites Heft, S. 128) wird auf die S. 36 von Reuschs Abhandlung [https://babel.hathitrust.org/cgi/pt?id=hvd.fl33pn;view=1up;seq=7 Der Spitzbogen] verwiesen.

{{blockquote|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:

{{blockquote|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:Coxeter, op. cit., S. 242DUDEN: Rechnen und Mathematik. 1985, S. 652

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

• {{MathWorld|title=Commandino's Theorem|urlname=CommandinosTheorem}}
• [http://pballew.blogspot.de/2012/09/a-couple-of-nice-extensions-of-median.html A Couple of Nice Extensions of the Median Properties]

Category:Theorems about polyhedron

Category:Theorems in geometry

Category:Euclidean geometry

Category:Tetrahedra