Given obtuse 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 or disprove:

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

which disproves 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

which disproves the second part.

Return to Assignment 8.