Journey to the Center of a Triangle

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 this question.

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.

Problem Statement:

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 problem statement.

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 problems.

A special note to Teachers

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 hints!

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 answers please!