Assignment Eight: Properties of the Orthocenter

By: Melissa Wilson

For this assignment, let us consider the triangle ABC below. Point H is the orthocenter of the triangle (formed by the altitudes of each side). Points D, E, F are the points where the feet of the altitudes meet each side.

Objective One -

Our first objective is to prove:

We will start by looking at the area of the whole triangle ABC:

We can also use the other sides as bases and find the other formulas for the area of triangle ABC:

By doing this we can see that

Now, we will look at the three smaller triangles in ABC. From the above picture, each smaller triangle is shaded a different color. We can say that the whole area of triangle ABC is the sum of the three smaller triangles:

Then we equate the above equation to the original area equation using the line segments AD and BC:

Clearing out the common factor of 1/2 gives:

We know that we want the left hand side of the equation to equal 1, so we will divide each side by

From before we know that , so substituting we arrive at,

Then, we can simplify all terms and we arrive at our objective.

Objective Two -

Now, we will prove:

We will use the previous result we proved,

We can also rewrite the segments using segment addition properties:

Plugging these new equations into the result from the previous objective, we arrive at:

Then, we can simplify and rearrange the equation,

This is the result we wanted to prove.

When the triangle becomes obtuse (one angle is larger than 90 degrees), then the orthocenter moves outside of the triangle, as shown below. We want to see if the two properties we just proved are still valid in this case.

First we will look at:

From the image we can see that and that leads to

Since we know this term is greater than one and the other two terms cannot be negative, we know that

This shows that this property does NOT hold for obtuse triangles.

Now, we will look at:

By similar analysis to the above argument, we can show that

Since we know two terms are larger than 2 and the third term cannot be negative, we arrive at

Therefore, this property also does NOT hold for obtuse triangles.