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:
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.