page_title: Incircle and excircles of a triangle
section_title: Gergonne triangle and point
hierarchical_section_title: Incircle and excircles of a triangle / Incircle and incenter / Gergonne triangle and point
context_page_description: In geometry, the incircle or inscribed circle of a triangle is the largest circle contained in the triangle; it touches the three sides. The center of the incircle is a triangle center called the triangle's incenter.
  An excircle or escribed circle of the triangle is a circle lying outside the triangle, tangent to one of its sides and tangent to the extensions of the other two. Every triangle has three distinct excircles, each tangent to one of the triangle's sides.
  The center of the incircle, called the incenter, can be found as the intersection of the three internal angle bisectors. The center of an excircle is the intersection of the internal bisector of one angle and the external bisectors of the other two. The center of this excircle is called the excenter relative to the vertex, or the excenter of. Because the internal bisector of an angle is perpendicular to its external bisector, it follows that the center of the incircle together with the three excircle centers form an orthocentric system.
  All regular polygons have incircles tangent to all sides, but not all polygons do; those that do are tangential polygons. See also Tangent lines to circles.
context_section_description: The Gergonne triangle (of ) is defined by the three touchpoints of the incircle on the three sides. The touchpoint opposite  is denoted , etc.
  This Gergonne triangle, , is also known as the contact triangle or intouch triangle of . Its area is 
  where , , and  are the area, radius of the incircle, and semiperimeter of the original triangle, and , , and  are the side lengths of the original triangle. This is the same area as that of the extouch triangle.
  The three lines ,  and  intersect in a single point called the Gergonne point, denoted as  (or triangle center X₇). The Gergonne point lies in the open orthocentroidal disk punctured at its own center, and can be any point therein.
  The Gergonne point of a triangle has a number of properties, including that it is the symmedian point of the Gergonne triangle.
  Trilinear coordinates for the vertices of the intouch triangle are given by
  Trilinear coordinates for the Gergonne point are given by 
  or, equivalently, by the Law of Sines,
caption: A triangle, 
    
      
        
          △
          A
          B
          C
        
      
      {\displaystyle \triangle ABC}
    
  , with      incircle,      incenter (
    
      
        
          I
        
      
      {\displaystyle I}
    
  ),      contact triangle (
    
      
        
          △
          
            T
            
              A
            
          
          
            T
            
              B
            
          
          
            T
            
              C
            
          
        
      
      {\displaystyle \triangle T_{A}T_{B}T_{C}}
    
  ) and      Gergonne point (
    
      
        
          
            G
            
              e
            
          
        
      
      {\displaystyle G_{e}}
    
  )