Triangles Formed by the Construction of the Orthocenter

Allyson Faircloth

For this activity, we will investigate the triangles formed by the construction of the orthocenter.  Refer to the picture below.

First we are going to prove that .

Proof:

First let the area of triangle ABC be K.  Then the area of triangle ABC must be equal to the sum of the areas of the six new triangles formed by the construction of the orthocenter.

Therefore,

                

However, we know that .

So, .

Then     .

Thus,   .

Next, we will prove that .

Proof:

Let's begin with what we previously proved.

          

           .

But we know that , and .

Therefore, .

                

But we know that .

So, .

              

              

Thus, .

 

Now we will look at the case where triangle ABC is obtuse. Refer to the picture below.

The above proofs will not generalize for an obtuse triangle ABD.

For an obtuse triangle, only one of the perpendicular lines HE will intersect with a side.  The other two (line AF and line HD) will intersect with the vertices A and C respectfully. Therefore, the orthocenter lies outside of triangle ABC.  For that reason, the area of the triangel ABC is split in half by the first perpendicular line HE, and the other triangle are outside of the triangle ABC forcing the sum of the areas of the triangles formed by the construction of the orthocenter to be greater than that of triangle ABC.

       

 

 

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