Orthocenters

by Zack Kroll

Given acute triangle ABC. Construct the Orthocenter H. Let points D, E, and F be the feet of the perpendiculars from A, B, and C respectfully. Our goal in this exploration is to use our knowledge of Orthocenters in order to prove two things:

and

We begin by looking at the ways in which we can find the area of triangle ABC. Because the lines used to create the Orthocenter are altitudes, we can determine the area of this triangle multiple ways. If we break it down into separate triangles we can examine them separately before combining them together into one, larger triangle.

As we know the formula for determining the area of a triangle is , but for the sake of eliminating fractions we can simply say that . The area of  can be represented in the following ways:

We can also represent the area of another way as well.

This equation can be rewritten by substituting equivalent expressions for 2A.

After dividing, we are left with the equation:

Now that we have proved the first part, we must use it in order to prove the second. We being with:

From here we can simplify this equation in order for it to represent what we want it to.

Therefore, we have proved that:

and