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 We can also represent the area of 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:
, but for the sake of eliminating fractions we can simply say
that 
. The area of 
can be
represented in the following ways:
another way as well.
and 