In this project, I will be looking at relationships between the various centers of triangles: incenter, circumcenter, centroid, and orthocenter.
I’ll be teaching geometry in the fall for the first time in a long, long time. I used to love it. I chose this project because I do not know (or remember) the relationships between the various centers or even what the centers themselves are. This project will give me a chance to review important vocabulary like the name for the segment connecting a vertex with the midpoint of its opposite side. Is it median? I would expect the centers would be things like the point where the altitudes of the triangles meet or the point that can be used as a center of a circle circumscribing the triangle.
I guess the place to begin is with definitions. The geometry text I will be using only had the definition for centroid. Using WolframAlpha I found definitions for the rest of them. I had to look quickly only for the definition and then close the page because WolframAlpha contains a plethora of information on each center and would spoil my fun. I’ll come back to the WolframAlpha pages at the end I’m sure. Below are the definitions for the various centers:
- incenter-intersection of the angle bisectors of a triangle
- circumcenter-intersection of the perpendicular bisectors of the sides of a triangle
- centroid-intersection of the medians of a triangle
- orthocenter-intersection of the altitudes of a triangle
To begin the exploration, for each of the centers, I:
- constructed a triangle by connecting three line segments at their endpoints.
- constructed the appropriate lines or line segments (angle bisectors, perpendicular bisectors, medians, and altitudes).
Below are the results of these constructions:
To construct the angle bisectors, I selected each pair of sides and then chose Angle Bisector under the Construct Menu.
To construct a perpendicular bisector, I selected a side and constructed the midpoint of the side. Then I selected the side and its midpoint and chose Perpendicular Line under the Construct Menu. I repeated this for the other two sides.
To construct a median, I selected a side and constructed its midpoint. Then using the segment tool, I constructed a segment from a vertex to the midpoint of the opposite side. I repeated this for the other two sides.
To construct an altitude, I selected a vertex and the side opposite the vertex and chose Perpendicular Line from the Construct Menu. Then I repeated this for the other two vertices.
Are the centers always within the triangles?
It appears that the various centers of the triangles are within the triangle. All of the above triangles are acute however. I am going to drag my triangles to produce acute, right, and obtuse triangles and see what happens to each of the centers. I know the centroid will be within the triangle but I am not sure about the other three.
Below are pictures and data that will help to answer the question.
The incenter and centroid will be in the interior of the triangle whether the triangle is acute, right, or obtuse. The circumcenter and orthocenter are in the interior of the triangle only when the triangle is acute. When the triangle is right, the circumcenter is located at the midpoint of the side opposite the right angle. The orthocenter of a right triangle is located at the vertex of the right angle; that is, when two of the sides, the legs, are altitudes of the triangle. The circumcenter and orthocenter of an obtuse triangle are located outside the triangle.
As I dragged vertex C to make angle ABC acute, right, or obtuse, I noticed that interesting things happened when the triangles were nearly equilateral. Below are some of these observations:
Altitudes are constructed from a vertex perpendicular to the opposite side. In equilateral triangles, the altitudes bisect the angles and bisect the opposite sides.
To form the circumcenter, perpendicular bisectors are constructed so they must bisect the sides of the triangle. In equilateral triangles they also bisect the angles.
After considering the orthocenters and circumcenters of equilateral triangles I went back to the definitions and realized that in equilateral triangles, the perpendicular bisectors and angle bisectors were the same. In fact, when you think about it, the altitudes, perpendicular bisectors, medians and angle bisectors all seem like they will be the same in equilateral triangles. Do the centers coincide in equilateral triangles?
I decided to construct an equilateral triangle and then construct all the centers and see if they coincided. First I constructed an equilateral triangle using two circles.
Then I hid the circles and constructed the angle bisectors, perpendicular bisectors, medians, and altitudes. I thought I’d take a vertex and its opposite side and construct all the different types for that pair. That way I could see whether they coincide.
So, for a particular vertex and opposite side, the angle bisector, perpendicular bisector, median, and altitude coincide. Without constructing the rest of them, I know the various centers coincide in equilateral triangles. With more time it would be good to consider how centers are related in other special triangles.
Center of What?
While I may not have time to complete all I would like to do, for the next part, I plan to measure various parts of the pictures to:
- see if I can determine anything about the location of each type of center relative to the sides, vertices, and angles of the triangle. For instance, is a particular center equidistant from the vertices of the triangle?
- see if there is anything special about the angles and segments at the center. For instance, when the angle bisectors intersect to form the incenter, each angle bisector is divided into two segments and there are many angles formed with the center as the vertex. Is there anything special about these angles and these segments?
The names, circumcenter and incenter, suggest that they can be used to circumscribe a circle about the triangle or inscribe a circle inside the triangle. I thought I’d start with these.
First the circumcenter…
The circumcenter is equidistant from the vertices of the triangle. Constructing a circle with the circumcenter as its center and the distance from the circumcenter to a vertex as a radius does construct a circle that circumscribes the triangle. Why do the perpendicular bisectors produce a point equidistant from the vertices? Consider the drawing below.
Now, the incenter…
I was surprised during the investigation of the incenter. At first I looked to see if the incenter was equidistant from the point where the angle bisectors intersected the sides of the triangle. It sure looked like it. However, when I measured these distances they were not the same. So then I measured the distance from the incenter to each of the sides. This distance is the same. Using the incenter as the center of the circle you can construct a circle that would inscribe the triangle. (I need to look at my construction again. I did it wrong. I used the incenter as the center of the circle and stretched the radius until it matched up with a point of intersection of an angle bisector and the opposite side. To identify a point where the circle would touch the triangle I would have to construct a perpendicular to a side from the incenter. Then I could center a circle at the incenter and drag to this point.)
The proof of this conjecture may come from looking at the two congruent triangles formed at each vertex. See triangle AIO and triangle AHO in the diagram below. They are congruent but I can’t see why without using circular reasoning. This will require more thought.
Extension - Circumcenters of Quadrilaterals
I think I remember that a circle can circumscribe a convex quadrilateral. Can the center of the circle be found by looking at the intersection of the perpendicular bisectors of the sides? Below is the constructions of the perpendicular bisectors of a quadrilateral. They do not intersect in a point. Looking at quadrilateral ABCD, it doesn’t seem like a circle could circumscribe it. I need to think about this more.
I am going to draw in a diagonal of a quadrilateral to split it into two triangles. Then I am going to get the circumcenter for one of the triangles and use it to circumscribe that triangle and see if it appears to circumscribe the quadrilateral.
This did not produce a circle circumscribing the quadrilateral. I can imagine a circle circumscribing certain quadrilaterals like rectangles, squares, and isosceles trapezoids. Others I can imagine not being able to circumscribe like non-square rhombi. What about other less special quadrilaterals? I would like to look into this more but do not have the time right now.
What I Learned
Below are a few of the specific mathematics insights I had while investigating the centers of triangles:
- The angle bisectors of a triangle intersect in one point as do the perpendicular bisectors, the medians, and the altitudes.
- In equilateral triangles, all the centers coincide.
- The circumcenter is equidistant from the vertices of the triangle and can be used to circumscribe the triangle.
- The incenter is equidistant from the sides of the triangle and can be used to inscribe the triangle.
- Some but not all quadrilaterals can be circumscribed.
Working through this investigation helped me improve my GSP skills. It gave me a chance to practice using a wide variety of GSP tools like constructing a perpendicular to a line at a point and tabulating measurements. Also, out of necessity, I learned efficient ways to do things like construct midpoints of all three sides at once. Learning how to create a picture from a GSP figure that can be imported into other documents is a skill that will help me when creating tests and activities.
Two more general insights I had while doing this investigation are:
- Do not jump to conclusions. I was surprised when looking at a circle inscribing a triangle and also when trying to circumscribe a quadrilateral.
- Investigating takes time but it is worth it.
Implications for the Classroom
Investigations like this require students to review definitions. The open-ended nature of it helps them work on their question-posing skills too. The direction their exploration takes is up to them. Students can take ownership of their discoveries. Everyone can notice something interesting and there is no limit to what they can discover. GSP makes a profound visual statement. For instance, the discovery that the angle bisectors intersect in one point as do the perpendicular bisectors, medians, and the altitudes is extremely clear. No one has to tell them or convince them. Seeing that no matter how you drag the triangle these points of intersection stay fixed is powerful.
To begin the project I needed to find out the definitions for the various centers of triangles. In the classroom I would probably have students begin with looking at angle bisectors, perpendicular bisectors, medians, and altitudes of triangles. I think it is surprising that each intersects in one point. They could construct a triangle and construct the angle bisectors and then notice no matter how they drag it the angle bisectors intersect. The names could come later. Knowing the names took away some of the discovery. The names suggested what the particular center would be good for. For instance, the name circumcenter suggested that it was a point that could be used as the center of a circle that would circumscribe a triangle.
One of the topics in the TEAM-Math Curriculum Guide for Geometry most directly related to this activity is to “identify special segments of triangles, their intersections and their measurement properties.” Several other content-oriented objectives were also reinforced such as proving triangles congruent and types of triangles. Several general topics from the guide were required to carry out the investigation including:
- analyze information in geometric contexts to see what relationships exist, and
- use inductive and deductive reasoning.
This investigation would make an excellent activity in a geometry classroom. It emphasizes reasoning and sense making. Each group could be assigned one of angle bisectors, perpendicular bisectors, medians, or altitudes to explore. That way they could focus in-depth on one type of center and then present their findings to the class. This would also help with time issues. Some groups may need help identifying areas to explore to begin with while others may need help focusing their efforts. I think this would be fun, satisfying, and very worthwhile!
Curriculum for geometry. (2006). TEAM-Math Curriculum Guide. Retrieved July 19, 2010 from http://team-math.net/
Larson, R., Boswell, L., & Stiff, L. (2003). Geometry: Concepts and skills. Evanston, IL: McDougall Littell Inc.
Weisstein, E. (n.d.). Circumcenter. Retrieved July 17, 2010, from http://mathworld.wolfram.com/Circumcenter.html
Weisstein, E. (n.d.). Incenter. Retrieved July 17, 2010, from http://mathworld.wolfram.com/Circumcenter.html
Weisstein, E.(n.d.). Orthocenter. Retrieved July 17, 2010, from http://mathworld.wolfram.com/Circumcenter.html