
Orthocenter of a Triangle
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 socalled 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.
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.
Other triangle topics
General
Perimeter / Area
Triangle types
Triangle centers
Congruence and Similarity
Solving triangles
Triangle quizzes and exercises
(C) 2011 Copyright Math Open Reference. All rights reserved

