Vector – Incenter and Centroid of Triangle

 

                       We will use the previous result to find the incenter and centroid of a triangle.  Before that,  let’s recall some convention:

 

 

                      For   ABC, usually we use lower case letter for the length of the corresponding side :

 

                                     

 

                      And we just follow this convention without specifying them explicitly over and over again.

 

 

 

Theorem (  Coordinate of Incenter )

                The coordinate of the incenter of   ABC ( denoted as I ) can be represented as

                                     =   +   +  

 

                                           

 

           Proof:

                                Recall for the bisector of an angle inside a triangle, we have

                                               :  = :  =c:b

                                Thus,

                                                       =   +  

 

                                       And

                                                       =   =

 

                                Furthermore,

                                                   :  = :  =  c :  = c(b+c) : ca

                                                                 = (b+c):a

                                                     =   +  

 

                                The rest of the work is to substitute the known result into it.

 

                                          =    +  

                                             =   +   +  

 

 

Theorem ( Coordinate of Centroid )

                  G is the centroid of  ABC.  Then

                                          =  (  +  +  )

 

                                 

 

                 Proof:

                                 In classical geometry, we have

                                            :  =1:1

                                            :  =2:1

                                Thus,

                                               =  +

                                                =  +

                                 =  +  (  +  )

                                          =  (  +  +  )