**By**

*Ronnachai Panapoi*

In this write-up, we had chance to investigate some
interesting properties of the orthocenter of a triangle.

First of all,
let’s review the definition of the orthocenter of a triangle.

*“The orthocenter of a triangle is the point at which the
three altitudes of the triangle meet.”*

We will explore some
properties of the orthocenter from the following problem.

*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 that *

To begin with,
we will verify the first relation. From the given problem we can draw the
picture to help us easier see the way to go to our goal.

Click HERE to see this figure in the GSP.

Let’s prove
the first proposition that claim that

*Proof:*

_{}

Since

_{}, _{}

_{}, _{}

_{}, _{}

Therefore,

It’s
not tough to prove the second proposition, when we use the first proposition in
our proof.

The second
proposition is claimed that

Look at the
figure below again.

We will see
that

_{}

Thus,

_{}

These cases we
proved above are under the condition that the triangle is an acute triangle.

*What if the triangle is an obtuse
triangle?*

Click HERE to explore where the orthocenter
is, when the triangle is an obtuse triangle.

*We will see that the orthocenter lies outside the triangle
if it’s the obtuse triangle.*