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Theorem and special cases

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

a 2 + b 2 + c 2 + d 2 = e 2 + f 2 + 4 g 2 {\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 g {\displaystyle g} has length 0. In addition the parallel sides are of equal length, hence Euler's theorem reduces to

2 a 2 + 2 b 2 = e 2 + f 2 {\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:

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

Dividing by 2 yields the Euler–Pythagoras theorem:

a 2 + b 2 = e 2 {\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

Alternative formulation and extensions

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 A B C D {\displaystyle ABCD} Euler introduced an additional point E {\displaystyle E} such that A B E D {\displaystyle ABED} forms a parallelogram and then the following equality holds:

| A B | 2 + | B C | 2 + | C D | 2 + | A D | 2 = | A C | 2 + | B D | 2 + | C E | 2 {\displaystyle |AB|^{2}+|BC|^{2}+|CD|^{2}+|AD|^{2}=|AC|^{2}+|BD|^{2}+|CE|^{2}}

The distance | C E | {\displaystyle |CE|} between the additional point E {\displaystyle E} and the point C {\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 | C E | 2 {\displaystyle |CE|^{2}} is correction term that needs to be added to the original equation of the parallelogram law.²

M {\displaystyle M} being the midpoint of A C {\displaystyle AC} yields | A C | | A M | = 2 {\displaystyle {\tfrac {|AC|}{|AM|}}=2} . Since N {\displaystyle N} is the midpoint of B D {\displaystyle BD} it is also the midpoint of A E {\displaystyle AE} , as A E {\displaystyle AE} and B D {\displaystyle BD} are both diagonals of the parallelogram A B E D {\displaystyle ABED} . This yields | A E | | A N | = 2 {\displaystyle {\tfrac {|AE|}{|AN|}}=2} and hence | A C | | A M | = | A E | | A N | {\displaystyle {\tfrac {|AC|}{|AM|}}={\tfrac {|AE|}{|AN|}}} . Therefore, it follows from the intercept theorem (and its converse) that C E {\displaystyle CE} and N M {\displaystyle NM} are parallel and | C E | 2 = ( 2 | N M | ) 2 = 4 | N M | 2 {\displaystyle |CE|^{2}=(2|NM|)^{2}=4|NM|^{2}} , which yields Euler's theorem.³

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 R n {\displaystyle \mathbb {R} ^{n}} connected by edges so that they form a cycle graph.⁴

Notes

• 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

External links

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

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

References

1. Lokenath Debnath: The Legacy of Leonhard Euler: A Tricentennial Tribute. World Scientific, 2010, ISBN 9781848165267, pp. 105–107 /wiki/ISBN_(identifier) ↩

2. Deanna Haunsperger, Stephen Kennedy: The Edge of the Universe: Celebrating Ten Years of Math Horizons. MAA, 2006, ISBN 9780883855553, pp. 137–139 /wiki/ISBN_(identifier) ↩

3. Deanna Haunsperger, Stephen Kennedy: The Edge of the Universe: Celebrating Ten Years of Math Horizons. MAA, 2006, ISBN 9780883855553, pp. 137–139 /wiki/ISBN_(identifier) ↩

4. Geoffrey A. Kandall: Euler's Theorem for Generalized Quadrilaterals. The College Mathematics Journal, Vol. 33, No. 5 (Nov., 2002), pp. 403–404 (JSTOR) https://www.jstor.org/stable/1559015 ↩