Assignment 8

Altitudes and Orthocenters

by Emily Bradley

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:

Proof 1: Begin by looking at the area of triangles.

Note that

The sum of the areas of the smaller traingles equals the area of the whole traingle.

Proof 2:The segments of the altitudesare equal to the altitude minus the other segment

So

By looking at the measurements in the digram, we can see the above holds true.

What if ABC is an obtuse triangle?

If it's obtuse then the orthocenter H falls outside of the vertex of the obtuse angle. In the below diagram A is obtuse, and H falls above it. Points E and F do not exist in this case. E cannot fall on segment AC and be collinear with BH because the altitude no longer intersects this segment. The same is true of point F on segment AB.

For these reasons the above proofs will not hold true in the obtuse case.