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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,