Centers of Triangles
[Key Idea 1, 4, 7]
Objective: To construct the four centers of a triangle: the centroid (medians), circumcenter
(perpendicular bisectors of the sides), incenter (angle bisectors), and orthocenter (altitudes).
1. Centroid
a. Open a new sketch.
Construct a triangle.
A
b. Select two sides, under the
Construct menu, choose
Midpoints. Connect the
midpoints to the opposite
vertex to construct the
medians.
E
D
centroid
B
F
C
c. Locate the point of
intersection. Label this
point the centroid. Verify that the third median goes through this point.
d. The centroid is called the center of mass of the triangle. If you were to cut out this
triangle, you could balance it on the head of a pin placed at the centroid.
2. Circumcenter
e. Using the same triangle as before, hide the three medians and the centroid point. (Do
not delete, we will need these later! The three midpoints should be left showing.)
f. Through the midpoints, construct perpendiculars to two sides.
A
E
D
F
B
circumcenter
C
g. Locate the point of intersection. Label this point the circumcenter. Verify that the third
perpendicular bisector goes through this point.
Schoaff, 2004
Centers of Triangle
p. 35
h. To see the why this is called the circumcenter, select the circumcenter and point A.
Under the Construct menu choose Circle by Center+Point. You will see that the
circumcenter is the center of the circle that circumscribes the triangle.
As you can see, the circumcenter may be outside the triangle.
3. Incenter
i. Using the same triangle as before, under the Edit menu, choose Undo Construct
Circle. Hide the three perpendicular bisectors, the circumcenter point, and the three
midpoints of the sides. (Do not delete. Only triangle ABC should be left.)
j. Select an angle by clicking
on the three points in order,
then under the Construct
menu choose Angle
Bisector. Repeat this on a
second angle.
k. Locate the point of
intersection. Label this
point the incenter. Verify
that the third angle bisector
goes through this point.
A
incenter
B
C
l.
To see the why
this is called the
J
incenter, select the
incenter and segment
AB. Under the
incenter
Construct menu choose
Perpendicular. Locate
the point where the
B perpendicular from the
incenter intersects
segment AB. (The
C
perpendicular line is the
dotted line in the figure
above. The point of intersection is labeled J in my construction.)
A
m. Select the incenter point and the intersection point, then under the Construct menu
choose Circle by Center+Point.
Schoaff, 2004
Centers of Triangle
p. 36
4. Orthocenter
n. Using the same triangle as before, under the Edit menu, choose Undo Construct
Circle, Undo Construct Intersection and Undo Construct Perpendicular. Hide the
three angle bisectors and the incenter point. (Do not delete. Only triangle ABC should
be left.)
o. Select vertex A and BC , the side opposite. Under the Construct menu, choose
Perpendicular. Repeat for vertex B and side AC .
p.
Locate the point of
intersection. Label this point the
orthocenter. Verify that the third
altitude goes through this point.
As you can see, the orthocenter
may be outside the triangle.
orthocenter
A
B
C
5. The Euler Segment
This exploration uses the triangle and centers constructed above. Go under the Display menu
and choose Show All Hidden. What you will see is a horrible mess.
a. Starting with all the lines sticking out from the triangle, select about three at a time and
hide them. Continue until there are no lines or rays left sticking out from the triangle.
b. Now select the three medians and the three midpoints of the sides. Hide them as well.
c. You should now have only the four centers and the triangle remaining. (See figure on
next page.)
To make them easier to see, relabel the orthocenter as OC, the incenter as IC, the
circumcenter as CC, and the centroid as CE.
Schoaff, 2004
Centers of Triangle
p. 37
d. Drag a vertex of your triangle. Which centers appear to be collinear? Are these centers
collinear for any triangle? (Measure the sides to help determine the types of triangles.)
orthocenter
A
incenter
centroid
B
C
circumcenter
Are there any triangles where all four centers appear to be collinear?
A
I CE
C
Are there any triangles where all four centers are the
same point?
e.
Connect the orthocenter OC and the circumcenter
CC of your triangle with a segment. This is called the
Euler Segment. Now drag around one vertex of the
triangle. What do you notice?
CC
OC
C
B
Schoaff, 2004
Centers of Triangle
p. 38
f. Draw a horizontal segment across the top of your sketch. Select this segment and the
top vertex of your triangle. Under the Edit menu, choose Merge Point to Segment.
g. Under the Edit menu, choose Action Button and then slide to the right to Animation.
A dialog box should appear. Select medium as the speed. Then click on the OK
button. The Animate Point button should appear on your sketch. (See figure below.)
A
L
M
OC
C
IC
CE
Animate Point
CC
B
Click on the Animate button and observe what happens. Click on the button again to
stop the animation.
Geometer's Sketchpad comes with a file labeled Triangles.gsp that has interactive sketches
involving the centers of the triangle and the Euler line.
6. Loci of Centers
h. Select point OC and under the Display menu choose Trace Intersection. Start the
animation again. What shape is traced?
i. Change the relationship between the triangle and the line, that is, if A is the point
animated, then change the position of B and/or C. Under Display, choose Erase
Traces. Start the animation again. What do you conjecture to be true?
j. What happens when segment BC is perpendicular to the animation line? (Just when
you thought you knew the answer, ....)
k. Try the locus of some of the other centers. Select OC and under Display menu, choose
Trace Intersection to switch the trace off. Now select another point to trace. Report
your results. Any surprises?
Schoaff, 2004
Centers of Triangle
p. 39