Vector – Circumcenter and Orthocenter of Triangle

 

Motivation:

                        The representation of circumcenter or orthocenter of a triangle by using  its three vertices is quite complex. And it also depends on whether this triangle is with 2-domensional or 3-dimensional system. We do not give solution here directly. Instead, we introduce the methods to tackle the problems and some basic properties.

 

                      In 2-dimensional system,  if  a line and a point outside the line is given, how do you find the perpendicular line equation that passes the point as shown below?

 

                                             

 

                       Since this is on 2-dimension system,  you can immediately use  as the normal vector of the perpendicular line.  With normal vector and a point on that line, the line equation can be determined easily:

                                    

                                                    (x-d₁, y-d₂) = 0

 

                       But this is not the case when you put the diagram into a 3 dimensional system – you need directional vector to determine the equation for a line with a point known; it can not be determined by “normal vector”. There are infinitely many normal vectors pointing to different directions for a line in 3-dimensional space. With this in mind, we just take a look at circumcenter and orthocenter of a triangle in 2-dimensional system.

 

Orthocenter of a Triangle

 

                  With 3 three points  A(a₁, a₂) ,  B(b₁, b₂), C(c₁, c₂),  we would like to find the orthocenter of       ABC.

                                 

 

                 The line equation for the altitude passing A is

                                        (b₁-c₁, b₂-c₂)  (x-a₁, y-a₂) = 0

 

 

 

               Similarly, the line equation for the altitude passing B is

                                       (a₁-c₁, a₂-c₂)  (x-b₁, y-b₂) = 0

 

               With the two equations, we can find the coordinate of orthocenter.

 

 

 

 

 

 

 

 

Circumcenter of a Triangle

                Circumcenter can be found via the intersection of two perpendicular bisectors of the sides of a triangle. 3 three points  A(a₁, a₂) ,  B(b₁, b₂), C(c₁, c₂) are given,  we would like to find the circumcenter.

 

                                     

 

                       Equation of perpendicular line of  :

                                      (b₁-c₁, b₂-c₂)  (x- ,  y -  ) = 0

                       Equation of perpendicular line of  :

                                      (b₁-a₁, b₂–a₂)  ( x-  , y -  ) = 0

 

                With the two equations, the circumcenter can be found.

 

 

 

 

Geometry Property of Inner Product

                       Previously,  we know that for any two vectors   and   , we have

                                           =   cos

                                  where  is the angle spanned by the two vectors.

 

                      And please notice that   cos  is also the projection length of  on .

 

 

                       Consider the diagram below,  D, E, F are on the line L that line L is perpendicular to  . 

 

                                            

 

               It is natural that we have

 

                            =  =

                                          

 

               With this in mind, it is easy to have the following properties.

 

Property ( Orthocenter ):

 

                     H is the orthocenter of   ABC. Then

 

                                           =  =

 

                       

 

Property ( Circumcenter )

                    Let O be the circumcenter of   ABC. Then

                            

                               = ½ 

                               = ½