Orthogonal Attributes
by Molly McKee

Given triangle ABC with orthocenter H, our goal is to prove:

Where height refers to a perpendicular line from one vertex to the opposite side.  This is also called the altitude of a triangle.  In the case of this triangle, we are given that H is the orthocenter of ABC.  Therefore, any line which passes through a vertex, through H, and is perpendicular to the opposite side will be an altitude.

First, remember that the area of a triangle can be found by the following formula:

When the altitudes are shown, we can see that ABC is broken up into three smaller triangles.

Now let’s find the area of the three smaller triangles:
If we look again at ABC, we see that AD, BE, and CF can all be used as heights.  So we can write the area of ABC in the following three ways:

If we add the areas of these three triangles together, the result will be the area of ABC:

Now divide both sides by ABC:

Remember that we can rewrite ABC three different ways.  So now we have:

After reduction we can see that: