Orthocentric

By: John Vereen

We
are going to prove that, given triangle ABC with points D, E, and F as feet of
the perpendiculars from A, B, and C, that we have the relationship:

First
of all, we are going to look at the areas of the three triangles color-coded
above: HAC, HAB, and HBC. We see
that their areas are as follows:

HAC = (1/2)(AC)(HE)

HAB = (1/2)(AB)(HF)

HBC = (1/2)(BC)(HD)

Then, we can represent the area of
ABC three different ways because we have three legs of different altitudes with
different bases.

ABC= (1/2)(AD)(BC) = (1/2)(AB)(CF) =
(1/2)(AC)(BE)

So, when we consider the ratios of
the three triangles over ABC, we get:

Using what we have proven above, we
will prove that

We can re-write the first equation we
proved above as follows:

Therefore,