Altitudes and Orthocenters
For this assignment, I will explore 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, respectively. Prove:
(1) and (2) .
Below is a GSP sketch of the triangle ABC.
First I will prove the first statement. Notice the triangle ABC may be split into three smaller triangles: Triangle HBC, Triangle HAC, and Triangle HAB. I will use the areas of the triangles to prove this statement. Remember that the definition of an orthocenter is the intersection of the three altitudes of the triangle. The altitudes may also be known as the heights of the triangle.
We know that:
Area of Triangle ABC = the Sum of the Areas of the Smaller Triangles.
Area of Triangle ABC =
If you divide both sides by the Area of Triangle ABC, you get:
The Area of Triangle ABC can be written in three different ways:, , or . I am going to use all three in the equation.
After simplifying, I get. Hence, I have proved the first part.
Now I will show that.
Notice that in first part, I showed that. I use that fact here.
Thus, I have shown that.
A couple questions that may arise are “Does this only hold for acute triangles? What about if the triangle is obtuse?”
Below is a GSP sketch of an obtuse triangle.
Notice that the orthocenter occurs outside of the triangle. In order to calculate the areas, I need to extend the legs of the triangle. Below is a new sketch.
Now I want to see if (1) and (2) . Looking at the triangle and calculating the lengths, I get:
So the first statement is untrue and likewise, the second statement is false.
By looking at the graph, one can tell that the first statement will be false because since, since and since and . So since one part of the sum is greater than 1 and the other two parts are greater than 0, the whole sum must be greater than 1, so the first statement is false. Because the first statement is false and we use it to prove the second statement, it also must be false.