a Remember the definition of each point and find them one at a time.
B
b Remember the definition of each point and find them one at a time.
C
c Remember the definition of each point and find them one at a time.
D
d Draw the three points on the same triangle. Next, draw the line that passes through two of the points determined. Is the third point also on this line?
A
aExample Solution:
B
bExample Solution:
C
cExample Solution:
D
dConjecture: The circumcenter, centroid, and orthocenter of a triangle are collinear.
Practice makes perfect
a Let's consider the acute triangle below.
Next, we will find the circumcenter, centroid, and orthocenter of â–³ ABC.
The circumcenter is the intersection point of the three perpendicular bisectors. In the graph below we find it.
The centroid is the intersection point of the three medians. We find it as follows.
The orthocenter is the intersection point of the three altitudes. We find it as shown in the figure below.
Let's graph the three points on the same triangle.
b Let's consider the obtuse triangle below.
As before, let's find the circumcenter, centroid, and orthocenter of â–³ ABC.
The circumcenter is the intersection point of the three perpendicular bisectors. In the graph below we find it.
The centroid is the intersection point of the three medians. We find it as follows.
The orthocenter is the intersection point of the three altitudes. We find it as shown in the figure below.
Finally, let's graph the three points on the same triangle.
c In this part, we will consider a right triangle.
One more time, we will find the circumcenter, centroid, and orthocenter of â–³ ABC.
The circumcenter is the intersection point of the three perpendicular bisectors. In the graph below we find it.
The centroid is the intersection point of the three medians. We find it as follows.
The orthocenter is the intersection point of the three altitudes. We find it as shown in the figure below.
Let's graph the three points on the same triangle.
d Before making a conjecture, let's analyze the locations of the circumcenter, centroid, and orthocenter of the triangles we used in Parts A, B, and C.
From the graph above, it seems like the circumcenter, centroid, and orthocenter are collinear. So, let's draw a line that passes through the circumcenter and the centroid.
As we can see, each line that passes through the circumcenter and centroid also passes through the orthocenter. Then, we can write the following conjecture.
Conjecture
The circumcenter, centroid, and orthocenter of a triangle are collinear.
Extra
About the Conjecture
The conjecture we just did is true; that is to say, these three points are collinear and the line that passes through them is called the Euler line.
