Orthocenter of a triangle - Math Open Reference

Orthocenter of a Triangle

From Greek: orthos - "straight, true, correct, regular"

The point where the three altitudes of a triangle intersect.
One of a triangle's points of concurrency.

Try this Drag the orange dots on any vertex to reshape the triangle. Notice the location of the orthocenter.

The altitude of a triangle (in the sense it used here) is a line which passes through a vertex of the triangle and is perpendicular to the opposite side. There are therefore three altitudes possible, one from each vertex. See Altitude definition.

It turns out that all three altitudes always intersect at the same point - the so-called orthocenter of the triangle.

The orthocenter is not always inside the triangle. If the triangle is obtuse, it will be outside. To make this happen the altitude lines have to be extended so they cross. Adjust the figure above and create a triangle where the orthocenter is outside the triangle. Follow each line and convince yourself that the three altitudes, when extended the right way, do in fact intersect at the orthocenter.

Summary of triangle centers

There are many types of triangle centers. Below are four of the most common.

Incenter		Located at intersection of the angle bisectors. See Triangle incenter definition
Circumcenter		Located at intersection of the perpendicular bisectors of the sides. See Triangle circumcenter definition
Centroid		Located at intersection of medians. See Centroid of a triangle
Orthocenter		Located at intersection of the altitudes of the triangle. See Orthocenter of a triangle

In the case of an equilateral triangle, all four of the above centers occur at the same point.

The Euler line - an interesting fact

It turns out that the orthocenter, centroid, and circumcenter of any triangle are collinear - that is, they always lie on the same straight line called the Euler line, named after its discoverer.
For more, and an interactive demonstration see Euler line definition.

Constructing the Orthocenter of a triangle

It is possible to construct the orthocenter of a triangle using a compass and straightedge. See Constructing the the Orthocenter of a triangle.

Summary of triangle centers

The Euler line - an interesting fact

Related triangle topics

General

Perimeter / Area

Triangle types

Triangle centers

Congruence and Similarity

Solving triangles

Triangle quizzes and exercises