Prove:
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.