page_title: Angle bisector theorem
section_title: Proof 3
hierarchical_section_title: Angle bisector theorem / Proofs / Proof 3
context_page_description: In geometry, the angle bisector theorem is concerned with the relative lengths of the two segments that a triangle's side is divided into by a line that bisects the opposite angle. It equates their relative lengths to the relative lengths of the other two sides of the triangle.
context_section_description: A quick proof can be obtained by looking at the ratio of the areas of the two triangles  and , which are created by the angle bisector in . Computing those areas twice using different formulas, that is  with base  and altitude  and  with sides ,  and their enclosed angle , will yield the desired result.
  Let  denote the height of the triangles on base  and  be half of the angle in . Then
  and
  yields
caption: α
          =
          
            
              
                
                  ∠
                  B
                  A
                  C
                
                2
              
            
          
          =
          ∠
          B
          A
          D
          =
          ∠
          C
          A
          D
        
      
      {\displaystyle \alpha ={\tfrac {\angle BAC}{2}}=\angle BAD=\angle CAD}