Euler's quadrilateral theorem

For a convex quadrilateral with sides ${\displaystyle a,b,c,d}$, diagonals ${\displaystyle e}$ and ${\displaystyle f}$, and ${\displaystyle g}$ being the line segment connecting the midpoints of the two diagonals, the following equations holds:

${\displaystyle a^{2}+b^{2}+c^{2}+d^{2}=e^{2}+f^{2}+4g^{2}}$

If the quadrilateral is a parallelogram, then the midpoints of the diagonals coincide so that the connecting line segment ${\displaystyle g}$ has length 0. In addition the parallel sides are of equal length, hence Euler's theorem reduces to

${\displaystyle 2a^{2}+2b^{2}=e^{2}+f^{2}}$

which is the parallelogram law.

If the quadrilateral is rectangle, then equation simplifies further since now the two diagonals are of equal length as well:

${\displaystyle 2a^{2}+2b^{2}=2e^{2}}$

Dividing by 2 yields the Euler–Pythagoras theorem:

${\displaystyle a^{2}+b^{2}=e^{2}}$

In other words, in the case of a rectangle the relation of the quadrilateral's sides and its diagonals is described by the Pythagorean theorem.[1]

Euler's theorem with parallelogram

Euler originally derived the theorem above as corollary from slightly different theorem that requires the introduction of an additional point, but provides more structural insight.

For a given convex quadrilateral ${\displaystyle ABCD}$ Euler introduced an additional point ${\displaystyle E}$ such that ${\displaystyle ABED}$ forms a parallelogram and then the following equality holds:

${\displaystyle |AB|^{2}+|BC|^{2}+|CD|^{2}+|AD|^{2}=|AC|^{2}+|BD|^{2}+|CE|^{2}}$

The distance ${\displaystyle |CE|}$ between the additional point ${\displaystyle E}$ and the point ${\displaystyle C}$ of the quadrilateral not being part of the parallelogram can be thought of measuring how much the quadrilateral deviates from a parallelogram and ${\displaystyle |CE|^{2}}$ is correction term that needs to be added to the original equation of the parallelogram law.[2]

${\displaystyle M}$ being the midpoint of ${\displaystyle AC}$ yields ${\displaystyle {\tfrac {|AC|}{|AM|}}=2}$. Since ${\displaystyle N}$ is the midpoint of ${\displaystyle BD}$ it is also the midpoint of ${\displaystyle AE}$, as ${\displaystyle AE}$ and ${\displaystyle BD}$ are both diagonals of the parallelogram ${\displaystyle ABED}$. This yields ${\displaystyle {\tfrac {|AE|}{|AN|}}=2}$ and hence ${\displaystyle {\tfrac {|AC|}{|AM|}}={\tfrac {|AE|}{|AN|}}}$. Therefore, it follows from the intercept theorem (and its converse) that ${\displaystyle CE}$ and ${\displaystyle NM}$ are parallel and ${\displaystyle |CE|^{2}=(2|NM|)^{2}=4|NM|^{2}}$, which yields Euler's theorem.[2]

Euler's theorem can be extended to a larger set of quadrilaterals, that includes crossed and nonplaner ones. It holds for so called generalized quadrilaterals, which simply consist of four arbitrary points in ${\displaystyle \mathbb {R} ^{n}}$ connected by edges so that they form a cycle graph.[3]

• Deanna Haunsperger, Stephen Kennedy: The Edge of the Universe: Celebrating Ten Years of Math Horizons. MAA, 2006, ISBN 9780883855553, pp. 137–139
• Lokenath Debnath: The Legacy of Leonhard Euler: A Tricentennial Tribute. World Scientific, 2010, ISBN 9781848165267, pp. 105–107
• C. Edward Sandifer: How Euler Did It. MAA, 2007, ISBN 9780883855638, pp. 33–36
• Geoffrey A. Kandall: Euler's Theorem for Generalized Quadrilaterals. The College Mathematics Journal, Vol. 33, No. 5 (Nov., 2002), pp. 403–404 (JSTOR)
• Dietmar Herrmann: Die antike Mathematik: Eine Geschichte der griechischen Mathematik, ihrer Probleme und Lösungen. Springer, 2013, ISBN 9783642376122, p. 418

Wikimedia Commons has media related to Euler's theorem for quadrilaterals.

• Weisstein, Eric W. "Quadrilateral". MathWorld.