Some Properties of the Orthocenter of a Triangle.

 

By

 

Ronnachai  Panapoi

 

 
                       

                       

                        In this write-up, we had chance to investigate some interesting properties of the orthocenter of a triangle.

 

          First of all, letís review the definition of the orthocenter of a triangle.

         

          ďThe orthocenter of a triangle is the point at which the three altitudes of the triangle meet.Ē

 

          We will explore some properties of the orthocenter from the following problem.

Given triangle ABC. Construct the Orthocenter H. Let points D, E, and F be the feet of the perpendiculars from A, B, and C respectfully.

         Prove that

 

          To begin with, we will verify the first relation. From the given problem we can draw the picture to help us easier see the way to go to our goal.

        

          Click HERE to see this figure in the GSP.

 

          Letís prove the first proposition that claim that

 

 

Proof:

 

 

Since  

                      , 

                      , 

                      ,  

 

Therefore,

         

                 

 

         

 

         

 

 

         

 

         

                   Itís not tough to prove the second proposition, when we use the first proposition in our proof.

 

          The second proposition is claimed that

 

 

 

         

          Look at the figure below again.

 

 

          We will see that

                   Thus,

 

         

         

          These cases we proved above are under the condition that the triangle is an acute triangle.

 

          What if the triangle is an obtuse triangle?

 

 

 

          Click HERE to explore where the orthocenter is, when the triangle is an obtuse triangle.

 

          We will see that the orthocenter lies outside the triangle if itís the obtuse triangle.

 

          RETURN