Median and Centroid of Triangle

 

Theorem:  If M is the Centroid of   ABC as below  ( D, E, F are the midpoints of each side respectively ), then 

               

                                            = 2

 

                                

 

              Proof:

 

                     We have introduced this property when we proved the existence of Centroid.

 

1.      From point F,  construct a parallel line to  as shown above.

      In other words,

                            // 

           =   ( F is the midpoint of   )

2.      Similarly, 

          =   = 2

                                   Thus,    = 2

 

 

Theorem:   M is the midpoint of    .  Then   +  > 2  .

 

                        

 

                 Proof:

1.      On , find a point D such that   =  as shown above.

2.       = ,   BMD =  CMA,  = .  Then

         AMC   BMD

       = 

3.      In   ABC, we have

        +   >

     So,      +  > 2 .

 

 

Theorem:   O is the centroid of    ABC.   Then

                                      BOC =  COA =  AOB  .

 

                            

                  Proof:

                              1.     =  2 .

                                     So,   BOA = 2  BOP.

                              2.    =  .  Thus,   BOP =  COP .

                              3.  From 1 and 2,  we have    AOB =  BOC .

                                   Similarly,   AOB =  COA.

                                    BOC =  COA =  AOB .