Orthocenter of a triangle

1 Star2 Stars3 Stars4 Stars5 Stars (No Ratings Yet)
Loading...

Drawing of the orthocenter of a triangle as the intersection of the three altitudes

In a triangle ABC the orthocenter H is the intersection point of the three altitudes of the triangle.

Every triangle has three altitudes (or heights) and three sides (or bases).

An altitude of a triangle (ha, hb y hc) is a perpendicular line segment from a vertex to the opposite side. This line containing the opposite side is called the extended base of the altitude.

Drawing of the outer orthocenter to the triangle and its three heights

Altitude can also be understood as the distance between the base and the vertex.

Where is the Orthocenter of a Triangle Located?

  • If it’s an obtuse triangle the orthocenter is located outside the triangle (as we see in the picture above).
  • If it’s an acute triangle the orthocenter is located inside the triangle.
  • If it’s a right triangle the orthocenter lies on the vertex of the right angle.

Euler Line

In any non-equilateral triangle the orthocenter (H), the centroid (G) and the circumcenter (O) are aligned. The line that contains these three points is called the Euler line.

Drawing of Euler's line

In an equilateral triangle all three centers are in the same place.

The relative distances between the triangle centers remain constant.

Distances between centers:

It is true that the distance from the orthocenter (H) to the centroid (G) is twice that of the centroid (G) to the circumcenter (O). Or put another way, the HG segment is twice the GO segment:

Formula for the relation of the distances between centers on the Euler line

When the triangle is equilateral, the barycenter, orthocenter, circumcenter, and incenter coincide in the same interior point, which is at the same distance from the three vertices.

This distance to the three vertices of an equilateral triangle is equal to Distance 1 on Euler's Line from one side and, therefore, Distance 1 on Euler's Line to the vertex, being h its altitude (or height).

Practice Exercise

Drawing in example 1

Find the coordinates of the orthocenter H of a triangle ABC with the vertex coordinates A (3, 5), B (4, -1), and C (-4, 1).

SOLUTION:

In order to find the orthocenter we have to solve the equations for the altitudes ha (starting from vertex A) and hb (from vertex B) and see where is located the intersection point (H) of both altitudes.

Drawing 2 in example 1

STEP 1: Find equation for ha:

Firstly we will find the equation of the line that passes through side BC, which is the opposite of vertex A. This equation is obtained knowing that it passes through points B (4, -1) and C (-4, 1). The general equation of the line that passes through two known points is:

Calculation of line BC in exercise 1

The equation of the line that contains side BC and its slope m will be:

Calculation of the line containing BC in exercise 1

The slope of the line that contains the altitude ha, being perpendicular to the side BC, is the inverse and of the opposite sign to the slope of the line that contains the side. So, we have that:

Calculation of the slope of the line in exercise 1

Knowing that the height ha passes through the vertex A (3, 5), we can obtain the equation of its line. The equation of a line, knowing a point and its slope (inverse slope and opposite sign to that found for BC, that is, mp = 4):

Calculation of the slope of the altitude ha in exercise 1

So, the equation for ha is y -4x +7 = 0.

STEP 2: Find equation for hb:

Now we proceed in the same way to find the equation of the line that contains the height hb, that is, to which, starting from vertex B, is perpendicular to side AC.

First, the slope of the line AC:

Calculation of the equation for the height ha in exercise 1

And, now, the equation of the line that contains the height hb, that is, the one that starts from vertex B and is perpendicular to side AC. Therefore, its slope mp will be –7/4.

Calculation of the slope of the line AC in exercise 1

Then, the equation of hb is 4y +7x -24 = 0.

We solve this system of two first degree equations with two unknowns. Its roots will be the coordinates of its intersection, that is, the location of the orthocenter H of the triangle ABC.

Calculation of the system of two equations in exercise 1

Solving this system of two first-degree equations we have that:

Calculation of the orthocenter solution in exercise 1

Therefore, the altitudes cross at (2.26, 2.04).

And the coordinates of the orthocenter are H (2.26, 2.04).

Drawing the solution of example 1 of the orthocenter of a triangle

AUTHOR: Bernat Requena Serra

YEAR: 2020


IF YOU LIKED IT, SHARE IT!

Leave a Reply

Your email address will not be published.