Students may need previous experience with specific ideas to reason
through the constructions in this lesson. For circumcenter, students
will need to understand that the perpendicular bisector of a chord
always passes through the center of the circle. For incenter, students
will need to know that, if the rays of an angle are tangent to a
circle, then the angle bisector passes through the center of the
circle.
This lesson begins with a hypothetical problem set in the real world. The problem can be stated as follows:
 Boise, ID; Helena, MT; and Salt Lake City,
UT are three large cities in the northwestern part of the United
States. Although each city has a local hospital for minor needs and
emergencies, an advanced medical facility is needed for transplants,
research, and so forth. Imagine the potential of a highpowered,
hightech, extremely modern medical center that could be shared by the
three cities and their surrounding communities! You have been hired to determine the best location for this facility.


To begin the lesson, project the Hospital Problem Overhead, which states the problem for students:
Hospital Problem Overhead
Use a map or atlas to show the location of the three cities. (You might want to have students access a map from an online map site, or you can project a copy of the Hospital Map Activity Sheet.)
Hospital Map Activity Sheet
Point out that there are no major towns located
between these cities that would be obvious locations for the medical
center; therefore, the major factor in determining a location should be
proximity to each city.
Distribute the Hospital Map Activity Sheet and a blank transparency sheet to all groups. Allow
students to work on the task in groups of two or three. Each group
should record their suggested solutions on a transparency sheet and be
ready to explain their thinking to the class. Students will often
create a triangle using the three cities as the vertices. Students may
use the Hospital Map Activity Sheet to trace the appropriate size triangle onto a transparency. The Cities Triangle Overhead can also be used, if students need it.
Cities Triangle Overhead
Students often suggest
that the medical center should be located "somewhere in the center" and
generally indicate a portion of the interior of the triangle. Pressing
them for a more specific idea of where "in the center" the medical
center should be placed may underscore the need for the group to
identify some criteria for selecting a "center" of the triangle.
Another common response from students is to want to locate a point that
is equidistant from each of the vertices. This suggestion—or a
conversation about "somewhere in the center"—leads to the discussion of finding a point that is equidistant from all of the vertices arises.
Use guiding questions to help students come to the conclusion that a circle needs to be circumscribed onto their triangle. This provides an opportunity to
introduce the terms circumcircle (a triangle's circumscribed circle) and circumcenter (the center of a triangle's circumcircle).
(Saving other locations
suggested by students may provide ways to introduce other special
points related to triangles.)
Have students experiment with trying to find the circumcircle and circumcenter, where the circumcenter is the location of the medical
facility. Provide blank activity sheets as needed to help students keep a record of how their work is being modified. The following interactive can be used to explore the circumcircle further.
Circumcircle
When students arrive at an answer, have them look at other
groups’ solutions. Then, conduct a discussion with the entire class
based on the following questions:
 What do you notice about all of the solutions?
[The point is in about the same place in each case.]
 How would we be able to construct the point we want if we were not allowed to use measurements?
[Students may not immediately see that the perpendicular
bisectors can be used, or they may mention them only because of what
appears on the screen. In any event, the point of the question is to
encourage students to wonder why the construction they are about to see
makes sense.]
 Why does it make sense that the circumcenter would lie on the perpendicular bisectors of the sides?
[Each side can be thought of as a chord of the circle. A radius of the circle must be a perpendicular bisector of a chord.]
 If "cutting the sides in half with perpendicular bisectors"
led to the circumcenter, what happens when we "cut the angle in half"?
[The angle bisectors intersect. That point is the center of a circle
that is tangent to all three sides. Discussion of the observations
includes introduction of the terms incircle and incenter to describe what students observe.]
After this discussion, have students explore the IncenterIncircle Tool (mobilefriendly). Explorations with this applet will lead to further discoveries by students. Use these discoveries to continue the discussion:
IncenterIncircle Tool
 Why does it make sense that the incenter would lie on the angle bisectors?
[Points on the angle bisector are equidistant from each of two sides of
the angle. This situation is necessary for each ray of the bisected
angle to be tangent to the inscribed circle.]
 How does "half of something" help us to make sense of how to construct circumcenters and incenters?
[Thinking "half of a side" in terms of perpendicular
bisectors leads us to perpendicular bisectors of chords of the
circumcircle. Thinking "half of an angle" suggests that angle bisectors
could be used to identify the incircle and to locate the incenter. From
half of a side or half of an angle, we get important ideas that
underlie how chords and tangents are related to the constructions.]
To wrap up the class, engage the class in discussing the mathematical answer of the hospital's location vs. its realworld plausibility. Students may realize that the answer found in the instruction is difficult to achieve due to the area's topography and geography (ex: the hospital would be too far away from any major highways).
Questions for Students
1. Why does it make sense that the circumcenter would lie on the perpendicular bisectors of the sides?
[Each side can be thought of as a chord of the circle. A radius of the circle must be a perpendicular bisector of a chord.]
2. Why does it make sense that the incenter would lie on the angle bisectors?
[Points on the angle bisector are equidistant from each of two sides of the angle. This situation is necessary for each ray of the bisected angle to be tangent to the inscribed circle.]
Teacher Reflection
 Where did students struggle in executing the constructions?
 How did students use their ideas about "half" to explain why the constructions for circumcenters and incenters make sense?