Assignment 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 respectively. Prove:

and also:

First, we draw a picture of the given triangle.

We can see that

and by substitution, we get

which leads to

and finally to

Now, we must remember the following:

Making the appropriate substitutions gives us

which reduces to

A little rearranging gives us

and we are done with the first part.

The second part depends on our result from the first part. It is helpful to look at our triangle again.

We begin with the equation:

We see from the drawing the following:

Substituting into our equation gives us

which results in

which simplifies to

Rearranging terms gives us

and we are done.

But, our triangle was acute.

Click here to see an exploration of an obtuse triangle.