Euler's theorem in geometry

In <a href="/facts/Geometry/wTQf5ffQ">geometry</a>, Euler's theorem states that the distance d between the <a href="/facts/Circumcenter/RafYT4am">circumcenter</a> and <a href="/facts/Incenter/e933kPfL">incenter</a> of a <a href="/facts/Triangle/cRXUwsMn">triangle</a> is given by

d
          
            2
          
        
        =
        R
        (
        R
        −
        2
        r
        )
      
    
    {\displaystyle d^{2}=R(R-2r)}

or equivalently

1
            
              R
              −
              d
            
          
        
        +
        
          
            1
            
              R
              +
              d
            
          
        
        =
        
          
            1
            r
          
        
        ,
      
    
    {\displaystyle {\frac {1}{R-d}}+{\frac {1}{R+d}}={\frac {1}{r}},}

where 
 
 
 
 R
 
 
 {\displaystyle R}
 
 and 
 
 
 
 r
 
 
 {\displaystyle r}
 
 denote the circumradius and inradius respectively (the radii of the <a href="/facts/Circumscribed_circle/RafYT4am">circumscribed circle</a> and <a href="/facts/Inscribed_circle/x9QCn9ML">inscribed circle</a> respectively). The theorem is named for <a href="/facts/Leonhard_Euler/z2fsbjgj">Leonhard Euler</a>, who published it in 1765. However, the same result was published earlier by <a href="/facts/William_Chapple_(surveyor)/hBU3IKE9">William Chapple</a> in 1746.
From the theorem follows the Euler inequality:

R
        ≥
        2
        r
        ,
      
    
    {\displaystyle R\geq 2r,}

which holds with equality only in the <a href="/facts/Equilateral_triangle/MJomsYDS">equilateral</a> case.

Euler's theorem in geometry open-in-new

Euler's theorem in geometry