Angle Bisector and Incenter of Triangle

 

Theorem:   As shown below,    is the bisector of   BAC.   Then

                                                  

                                                           

                                             

 

             Proof:

 

1.      From point C, draw a line such that the line is parallel to   

as shown above.  In other words,

 

              //   .

Thus,     BAD =  E  and   DAC =  ACE .

 

                          2.  Furthermore,    is the bisector of   BAC

                                  BAD =  DAC

 

                                Hence,    E =   ACE .

                                  = 

 

                          3.    // 

                               

 

                              Finally, we have

 

                                            

 

 

 

 

 

 

 

 

 

Theorem :   O is the incenter of   ABC. 

                     Then O is equidistant to the three sides of   ABC.

                     

 

                     Proof: 

                                  It can be shown by repeatedly using AAS congruence theorem.

 

Corollary:   O is also the center of the inscribed circle of     ABC. 

                     Then each side of the triangle is an tangent line of the circle.

                    This circle is also known as “Incircle” .

 

 

                  

 

                   Proof:

 

1.      From the theorem, we know that O is equidistant to three sides. Thus, a circle exists with center O and the distance to one side as radius.

 

2.      The three sides are perpendicular to its corresponding line segment of radius and  the crossing point is on the other end of the radius.  Thus, these three sides are tangent lines of the circle.

 

 

 

Corollary:  Let  r be the radius of the incircle of   ABC.   Then

                          ABC =  ½  (  +   +   ) r   

 

                   Proof:    It can be easily shown by using the diagram above of incircle.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Theorem:   In the following diagram,  O is the incenter of   ABC.

                         

                   Then

                                      = 1

 

               Proof:

 

                             1.    = , 

                                    =  ,

                                    =

                             2,    = 1  ( by using the equalities above ).