Write-Up #8

Altitudes and Orthocenters

Given triangle ABC.  Construct the orthocenter H.  Let points D, E, and F be the feet of the perpendiculars from A, B, and C respectfully.
Prove:


First, construct the orthocenter H of triangle ABC.

 

Next, let points D, E, and F be the feet of the perpendiculars from A, B, and C respectfully.

 

From the picture below, we can see that triangle ABC is composed of 3 smaller triangles, .

 

 

Since the 3 smaller triangles compose triangle ABC, we can represent the area of triangle ABC in terms of the areas of the 3 smaller triangles.

In other words, .

We can divide both sides by .

The area of a triangle is 1/2 * base * height. Substitute this formula into each of the triangles in our above equation.

Canceling factors in the numerators and denominators of each term, we find:

Rearranging terms:

Q.E.D.


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