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

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This can be rewritten as**

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Remember that we have proven that
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**So
it basically comes down to 3 – 1 = 2.**