Euler's quadrilateral theorem

<h2 id="theorem-and-special-cases">Theorem and special cases</h2>
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 <a href="/facts/Parallelogram/DRuktq60">parallelogram</a>, 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 <a href="/facts/Rectangle/mdLfIGKA">rectangle</a>, 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.<a class="footnote-ref" id="fnref:1" href="#fn:1">1</a>

<h2 id="alternative-formulation-and-extensions">Alternative formulation and extensions</h2>

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.<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a>

 
 
 
 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 <a href="/facts/Intercept_theorem/zrBsVL2P">intercept theorem</a> (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.<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a>
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 <a href="/facts/Cycle_graph/bYhnPDHc">cycle graph</a>.<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a>

<h2 id="notes">Notes</h2>

<ul><li>Deanna Haunsperger, Stephen Kennedy: The Edge of the Universe: Celebrating Ten Years of Math Horizons. MAA, 2006, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 9780883855553, pp. <a href="https://books.google.com/books?id=I9oVP8TlyqIC&pg=PA137">137–139</a></li>
<li>Lokenath Debnath: The Legacy of Leonhard Euler: A Tricentennial Tribute. World Scientific, 2010, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 9781848165267, pp. <a href="https://books.google.com/books?id=dDfICgAAQBAJ&pg=PA105">105–107</a></li>
<li>C. Edward Sandifer: How Euler Did It. MAA, 2007, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 9780883855638, pp. <a href="https://books.google.com/books?id=sohHs7ExOsYC&pg=PA33">33–36</a></li>
<li>Geoffrey A. Kandall: Euler's Theorem for Generalized Quadrilaterals. The College Mathematics Journal, Vol. 33, No. 5 (Nov., 2002), pp. 403–404 (<a href="https://www.jstor.org/stable/1559015">JSTOR</a>)</li>
<li>Dietmar Herrmann: Die antike Mathematik: Eine Geschichte der griechischen Mathematik, ihrer Probleme und Lösungen. Springer, 2013, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 9783642376122, p. <a href="https://books.google.com/books?id=CDgiBAAAQBAJ&pg=PA418">418</a></li></ul>
<h2 id="external-links">External links</h2>

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

<ul><li><a href="/facts/Eric_W._Weisstein/FbMuVDep">Weisstein, Eric W.</a> <a href="https://mathworld.wolfram.com/Quadrilateral.html">"Quadrilateral"</a>. <a href="/facts/MathWorld/yc1jodbq">MathWorld</a>.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Lokenath Debnath: The Legacy of Leonhard Euler: A Tricentennial Tribute. World Scientific, 2010, ISBN 9781848165267, pp. 105–107 <a href="/wiki/ISBN_(identifier)" target="_blank">/wiki/ISBN_(identifier)</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Deanna Haunsperger, Stephen Kennedy: The Edge of the Universe: Celebrating Ten Years of Math Horizons. MAA, 2006, ISBN 9780883855553, pp. 137–139 <a href="/wiki/ISBN_(identifier)" target="_blank">/wiki/ISBN_(identifier)</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Deanna Haunsperger, Stephen Kennedy: The Edge of the Universe: Celebrating Ten Years of Math Horizons. MAA, 2006, ISBN 9780883855553, pp. 137–139 <a href="/wiki/ISBN_(identifier)" target="_blank">/wiki/ISBN_(identifier)</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Geoffrey A. Kandall: Euler's Theorem for Generalized Quadrilaterals. The College Mathematics Journal, Vol. 33, No. 5 (Nov., 2002), pp. 403–404 (JSTOR) <a href="https://www.jstor.org/stable/1559015" target="_blank">https://www.jstor.org/stable/1559015</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
</ol>

Euler's quadrilateral theorem open-in-new

Euler's quadrilateral theorem