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 at 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.ref>Norman Schaumberger: Commandino's theorem. The Two-Year College Mathematics Journal, Vol. 13, No. 5 (Nov., 1982), p. 331 (JSTOR)</ref>

The theorem is attributed to Commandino, who stated, 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 to 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 Vinci's works. Other scholars have speculated that the result may have already been known to Greek mathematicians during antiquity.

External links

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