page_title: Fuhrmann triangle
hierarchical_section_title: Fuhrmann triangle
context_page_description: The Fuhrmann triangle, named after Wilhelm Fuhrmann, is special triangle based on a given arbitrary triangle.
  For a given triangle and its circumcircle the midpoints of the arcs over triangle sides are denoted by. Those midpoints get reflected at the associated triangle sides yielding the points, which forms the Fuhrmann triangle.
  The circumcircle of Fuhrmann triangle is the Fuhrmann circle. Furthermore the Furhmann triangle is similar to the triangle formed by the mid arc points, that is. For the area of the Fuhrmann triangle the following formula holds:
  Where denotes the circumcenter of the given triangle and its radius as well as denoting the incenter and its radius. Due to Euler's theorem one also has. The following equations hold for the sides of the Fuhrmann triangle:
  Where denote the sides of the given triangle and the sides of the Fuhrmann triangle.
context_section_description: The Fuhrmann triangle, named after Wilhelm Fuhrmann (1833–1904), is special triangle based on a given arbitrary triangle.
  For a given triangle  and its circumcircle the midpoints of the arcs over triangle sides are denoted by  . Those midpoints get reflected at the associated triangle sides yielding the points , which forms the Fuhrmann triangle.
  The circumcircle of Fuhrmann triangle is the Fuhrmann circle. Furthermore the Furhmann triangle is similar to the triangle formed by the mid arc points, that is . For the area of the Fuhrmann triangle the following formula holds:
  Where  denotes the circumcenter of the given triangle  and   its radius as well as  denoting the incenter and  its radius. Due to Euler's theorem one also has  . The following equations hold for the sides of the Fuhrmann triangle:
  Where  denote the sides of the given triangle  and  the sides of the Fuhrmann triangle (see drawing).
caption: Fuhrmann triangle (red): 
    
      
        
          △
          
            M
            
              c
            
            
              ′
            
          
          
            M
            
              b
            
            
              ′
            
          
          
            M
            
              a
            
            
              ′
            
          
        
      
      {\displaystyle \triangle M_{c}^{\prime }M_{b}^{\prime }M_{a}^{\prime }}
    
   
   mid arc points: 
    
      
        
          
            M
            
              a
            
          
          ,
          
            M
            
              b
            
          
          ,
          
            M
            
              c
            
          
        
      
      {\displaystyle M_{a},M_{b},M_{c}}