Assignment 8

by

Johnie Forsythe

Altitudes and Orthocenters

Let's explore some relationships between altitudes and orthocenters.

What is an orthocenter and an altitude?

The orthocenter of a triangle is the common intersection 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) The orthocenter (H) does not have to be on the segments that are altitudes. Rather, H lies on the lines extended along the altitude.

We can construct the orthocenter of a given triangle ABC using GSP.

The altitudes of a triangle divides the triangle into three separate triangles. Let's explore the relationship between the orthocenters of these interior triangles.

All of the interior triangles are obtuse triangles. The orthocenter of an interior triangle of the larger angle is coincident with the vertex of the larger triangle that is not on the interior triangle.

Where is the orthocenter of a right triangle?

The orthocenter of a right triangle is coincident with the vertex opposite the hypotenuse.

Now, let's construct the circumcircles of triangles ABC, AHC, BHC, and BHA.

All of the circumcircles are congruent. Their radi are equal.

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