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:
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.