Journey to the Center of a
What is the center of a triangle? This question might be more
complicated than you first think it is! My high school geometry
students in Honolulu explore this question during a three-day project
in which we explore this question using interactive geometry
software. The students work in small, collaborative groups exploring
triangles and various ideas they have for what might be the answer to
These web pages include some questions and possible solutions that
my students discovered, some hints, the answers, and some fascinating
applications of the center of a triangle.
If you would like to participate in this process of discovery,
then begin with the problem statement below.
What is the center of a triangle?
Think about some possible answers to this question before you go
any further. There are not necessarily any "right" or immediate
answers to this preliminary question. You may give an answer or
answers, pose questions related to the problem, sketch ideas, or
whatever you feel is appropriate, before we go on to the rest of the
Actually, there are a number of different answers to this
question. It really depends on what you mean by center! Why would you
want to find the center of a triangle? Well, let's look at 3
situations where you might need this information.
1) You are a city planner. The three towns of Kaneohe, Kailua and
Maunawili have pooled their funds and want to build a recreation
center. The 3 towns are shown in the map below. Where would you put
the rec center so as to be fair to all 3 towns?
2) You are a sculptor and have just completed a large metal mobile.
You want to hang this mobile, made of flat triangular metal plates,
in the State Capitol. Each triangular piece will hang so that it will
be suspended with the triangular surface parallel to the ground. From
what point should each piece hang?
3) You are an architect. You have been asked to design an addition to
a house for a client.Your client wants the addition to be circular.
The site on which it is to be built is given. The house must fit
within the setback lines (dashed lines) and be attached to the
existing house. Where would you place the center of the addition, to
make it as large as possible?
For each question, sketch some ideas, and think about possible
answers to the questions. If you have Geometer's Sketchpad, construct
a simple model of each situation, using points and segments, and
experiment with your points, segments, and triangles to look for some
solutions. If you don't have computer software, use pencil and paper,
or a compass and straightedge and explore constructions. Here's a
hint: think about each of these questions in terms of the point or
set of points that meets the conditions described.Here's another
hint: consider the locus of points that meets the required
conditions, and think of these questions as compound locus
A special note to
My students, in their explorations, came up with some conjectures
as to what the answers might be. The link below will take you to a
page of their comments and ideas.
Don't give me any hints yet;
just show me some student ideas
Or, if you have thought about these 3 questions enough, and
sketched some ideas of your own, perhaps you would like some hints:
click on the link below:
Please, show me some
If you are ready to jump straight to the answers to these 3
questions, click on the link below:
I give up; just give me the