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: