Salinon

<h2 id="construction">Construction</h2>
<p>Let <i>A</i>, <i>D</i>, <i>E</i>, and <i>B</i> be four points on a line in the plane, in that order, with <i>AD</i> = <i>EB</i>. Let <i>O</i> be the bisector of segment <i>AB</i> (and of <i>DE</i>). Draw semicircles above line <i>AB</i> with <a href="/facts/Diameter/6uNn4VmU">diameters</a> <i>AB</i>, <i>AD</i>, and <i>EB</i>, and another semicircle  below with diameter <i>DE</i>. A salinon is the figure bounded by these four semicircles.<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a>
</p>
<h2 id="properties">Properties</h2>
<h3>Area</h3>
<p>Archimedes introduced the salinon in his <i>Book of Lemmas</i> by applying Book II, Proposition 10 of <a href="/facts/Euclid%27s_Elements/nPKvCIxs">Euclid's <i>Elements</i></a>. Archimedes noted that "the area of the figure bounded by the circumferences of all the semicircles [is] equal to the area of the circle on CF as diameter."<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a>
</p><p>Namely, if 
  
    
      
        
          r
          
            1
          
        
      
    
    {\displaystyle r_{1}}
  
 is the radius of large enclosing semicircle, and 
  
    
      
        
          r
          
            2
          
        
      
    
    {\displaystyle r_{2}}
  
 is the radius of the small central semicircle, then the area of the salinon is:<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a>

A
        =
        
          
            1
            4
          
        
        π
        
          
            (
            
              
                r
                
                  1
                
              
              +
              
                r
                
                  2
                
              
            
            )
          
          
            2
          
        
        .
      
    
    {\displaystyle A={\frac {1}{4}}\pi \left(r_{1}+r_{2}\right)^{2}.}

</p>
<h3>Arbelos</h3>
<p>Should points <i>D</i> and <i>E</i> converge with <i>O</i>, it would form an <a href="/facts/Arbelos/1fe5HCuA">arbelos</a>, another one of Archimedes' creations, with <a href="/facts/Symmetry/hpu7DK8p">symmetry</a> along the <a href="/facts/Y-axis/lkTqvMmB"><i>y</i>-axis</a>.<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a>
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Lune_of_Hippocrates/1qPpVffU">Lune of Hippocrates</a></li></ul>

<h2 id="external-links">External links</h2>
<ul><li><a href="http://images.math.cnrs.fr/L-arbelos-Partie-II.html">L’arbelos. Partie II</a> by Hamza Khelif at <a href="http://images.math.cnrs.fr/">www.images.math.cnrs.fr</a> of <a href="/facts/Centre_national_de_la_recherche_scientifique/z8RBcY1j">CNRS</a></li></ul>

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

<ol>
<li id="fn:1"><p>Heath, T. L. (1897). "On the Salinon of Archimedes". The Journal of Philology. 25 (50): 161–163. <a href="/wiki/Thomas_Heath_(classicist)" target="_blank">/wiki/Thomas_Heath_(classicist)</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Nelsen, Roger B. (April 2002). "Proof without words: The area of a salinon". Mathematics Magazine. 75 (2): 130. doi:10.2307/3219147. JSTOR 3219147. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Bogomolny, Alexander. "Salinon: From Archimedes' Book of Lemmas". Cut-the-knot. Retrieved 2008-04-15. <a href="/wiki/Alexander_Bogomolny" target="_blank">/wiki/Alexander_Bogomolny</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>Weisstein, Eric W. "Salinon". MathWorld. <a href="/wiki/Eric_W._Weisstein" target="_blank">/wiki/Eric_W._Weisstein</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
<li id="fn:5"><p>Bogomolny, Alexander. "Salinon: From Archimedes' Book of Lemmas". Cut-the-knot. Retrieved 2008-04-15. <a href="/wiki/Alexander_Bogomolny" target="_blank">/wiki/Alexander_Bogomolny</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li>
</ol>

Salinon open-in-new

Salinon