Given acute 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:

The orthocenter H is created by connecting the three altitudes of triangle ABC.

Recall, that the altitude of a triangle is also the height of a triangle. Since the altitude of triangle ABC has been created, it follows that the area of this triangle, S, can be found by using the area formula, .

Now, the area of triangle ABC = where BC is the base and AD is the height.

The area of triangle ABC can also be found by the following

, where AB is the base and CF is the height.

, where AC is the base and BE is the height.

Thus,

There are several other triangles created by the altitudes. The sum of the areas of triangles AHB, BHC, and CHA are equivalent to the area of triangle ABC.

Therefore,

Prove:

Proving this equation requires the use of the equation that was just proved.

Now, it follows that the equation can be re-expressed as

By simplifying,

What happens when the triangle is no longer acute? What happens when it is an obtuse triangle?

To answer this question, it is necessary to think about what happens to the orthocenter when the triangle contains an obtuse angle.

For instance,

When the triangle is obtuse, the orthocenter is actually located outside of the circle. Additionally, there are only 2 new triangles created. These triangles share the altitude of the triangle ABC. Thus, the aforementioned equations do not hold.