Orthocenters

by

Rob Walsh

The ORTHOCENTER of a triangle is the common intersection point of the three lines containing the altitudes. An altitude is a perpendicular segment from a vertex to the line of the opposite side. (Note: the foot of the perpendicular may be on the extension of the side of the triangle.) It should be clear that the orthocenter does not have to be on the segments that are the altitudes. Rather, it lies on the lines extended along the altitudes.

First, we construct the orthocenter of a triangle as follows:

1. Define a triangle by three noncollinear points a, b, and c. Draw lines ab, bc and ac.

2. Draw perpendiculars from each line through its opposite vertex.

3. The intersection of these three perpendiculars are the orthocenter of the triangle. Consider its positioning relative to the triangle type.

In the above examples, we find that the orthocenter's location depends on the type of triangle it belongs to. Let's look at a gsp sketch for a more indepth look at the construction and path of an orthocenter. We gain further insight by use of a script tool to find a triangle's orthocenter, by contrasting triangle types and orthocenter locations and by tracing the path of an orthocenter as it moves in relation to the movement of a triangle.