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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