Assignment 8 Write-up

Billy Poe

12. 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:

1. 

2. 

At first, I thought that I could prove this by showing proportions, such as HD = 2HA.  However, after calculating these lengths I realized that this was not true.  (I must have been thinking about properties dealing with medians and centroids.)  Then I realized that I'm dealing with a whole lot of triangles, so what better way to prove this than by using a lot of properties of triangles?  First, let's remember that the area of a triangle = .5(base)(height).  Now, to get started. . .

Now, to prove the second equation. . .

The first thing to notice is that AD = AH + HD, BE = BH + HE, and CF = CH + HF.  Now, with a little switching and substituting, we get

 

This can be rewritten as

Remember that we have proven that

So it basically comes down to 3 1 = 2.

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