Assignment 8 - Altitudes and Orthocenters
Faith Hoyt
We are given triangle ABC. Next, we were instructed to construct the orthocenter, H, of our triangle. Let points D, E, and F be the feet of the perpendiculars from A, B, and C respectfully, as we use altitudes to find the orthocenter. We want to prove:
Here is what our beginning triangle should look like:
First, we will prove that the first sum of ratios will equal 1, or that 
.
In order to do this, we need to look at this triangle in terms of it's areas.
Now, we want to prove that the sum of the ratio of parts of the altitudes will equal 2. In other words, 
.
Now, the question is asked, what happens if triangle ABC is obtuse (click here for the GSP file to investigate on your own) ?
If we take ABC and make it obtuse, we would get the following picture. Notice by the calculations done to the side that the ratios do not hold true for an obtuse triangle. Rather, we get a much larger sum. The same is true for the lengths of the segments. We get a sum of about 3.3 rather than 2. Thus, neither of these ratio sums holds true for an obtuse triangle.
