Euler Line

 

Theorem ( Euler Line ) :  Let O be the Circumcenter, M be the Centroid, and

                                           H be the Orthocenter of   ABC.

                                           Then  O, M, H lie on one straight line.

                                      

 

               Proof:

 

                               Euler original proved this theorem by using “brutal force” to

                               find the coordinate of  each point – a generic approach being

                               used in analytic geometry.  Here, we introduce a “classical”

                               way to prove this theorem.

 

1.      Construct  and let’s denote the crossing point of   and 

                                as P;  D is the midpoint of    .

Since O is the Circumcenter of the triangle, then    .

 

2.      From C,  construct a line perpendicular to  and the line crosses   at point S as shown in the diagram above.

      Construct  and   .

 

3.      Let’s denote E as the midpoint of  .  Then     ( O is Circumcenter ).

4.        //  and D is the midpoint of 

         = 2  and   = .

5.    =  and  E as the midpoint of 

         // 

           

6.   Look at  C, H, A, S :     // , and   //

         = 

 

       Thus,    =   = 2  ( from 4 )

7.     // 

        OPD   HPA

         

 

        And from 6,  we know that  

              

 

       Furthermore,  P is on  .  So, we know that P = M, i.e,  P is the Centroid of the triangle.

   Recall that P is the intersection of   and  . Hence  O, M, H lie on the same line.