Commandino's theorem

medians of a tetrahedron intersecting in centroid S

${\begin{aligned}&{\frac {|AS|}{|SS_{BCD}|}}={\frac {|BS|}{|SS_{ACD}|}}={\frac {|CS|}{|SS_{ABD}|}}\\=&{\frac {|DS|}{|SS_{BBC}|}}={\frac {3}{1}}\end{aligned}}$

Commandino's theorem, named after Federico Commandino (1509 – 1575), states that the four medians of a tetrahedron are concurrent in a point S, which divides them in a 3:1 ratio. Note that 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 described in his work De Centro Gravitates Solidorum (The Center of Gravity of Solids, 1565) that the four medians of the tetrahedron are concurrent. However, according the 19th century scholar G. 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 Vincis's works. Some other scholars speculated that the result might have been known already to Greek mathematician during antiquity.

External links

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