Assignment 4
Centers
of a Triangle
by Rives Poe
In this investigation we
are going to explore the relationships of the centers of triangles.
The first center to look
at is the CENTROID. (HINT: The centroid is always labeled
G. It is labeled G, because there is more than one center so some
mathematicians had to get a little creative and it also stands
for the center of Gravity of the triangle.)
The centroid of
a triangle is the common intersection of the three medians. A
median of a triangle is the segment from a vertex to the midpoint
of the opposite side. Below is a construction of the centroid.
Click here to see the construction and change the shape and
size of the triangle.
The next center we can
look at is the ORTHOCENTER. (HINT: The orthocenter
is always labeled H.) 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. Below is a construction
of the ORTHOCENTER of a triangle.
Click here
to explore the orthocenter on GSP with various shaped
and sized triangles.
Next we will explore the
CIRCUMCENTER of a triangle (always labeled C -finally!) The circumcenter of
a triangle is the point in the plane equidistant from the three
vertices of the triangle. Since a point equidistant from two points
lies on the perpendicular bisector of the segment determined by
the two points, C is on the perpendicular bisector of each
side of the triangle. Look below to see the orthocenter of a triangle.
Notice the circle around the triangle. This is called the circumcircle.
It is the circumscribed circle of the triangle, meaning a segment
from the circumcenter of the triangle to a vertex creates the
radius of the circle.
Click here
to use GSP to explore the
cirumcenter of a triangle.
Now that we have seen the
centers of the triangle, look at them all together. Do you notice
any relationship between the points? Why do you think they line
up like they do?
The centroid G is in the
middle of the circumcenter and orthocenter. Do you think that
for any triangle they will form a straight line? Will the line
always be inside the triangle? Click here
to investigate further on GSP.
Let's take a look at the
triangle below. Look at the ratio between segment HG and segment
GC.
Notice the triangle formed
by A',B', and D'. This triangle is called the medial triangle,
because the vertices of the triangle stem from the midpoints of
triangle ABD. Therefore, the lengths of triangle A'B'D' sides
are half the length of the sides of triangle ABD. The centroid
(G) is the same for both triangle. The orthocenter for A'B'D'
is actually the circumcenter for triangle ABD. The circumcenter
for A'B'D' does not fall on a point on the Euler Line for ABD.
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