Orthogonal Attributes

by Molly McKee

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

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 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: