Ma668, Fall 1997

For: Dr. J. Wilson

By: Sandra McAdams

In this exploration we will examine the relationships between inscribed
polygons and circles. The exploration can begin without the aid of technology
with simple paper folding. The question posed, "Can a circle be drawn
through the vertices of any triangle?" Ask the students to do the following:

1.Draw an acute scalene triangle on a piece of paper and label it ABC.

2. Fold the paper so that A lies directly on top of C. Crease the paper
and determine if there is anything special about this line?

3.Continue the activity by placing B on top of C and A on top of point B.
Each time the paper crease will yield a special line. What line is this?

4. Look at the intersection of the three lines. Label this point O. What
characteristics would be true of point O with regard to its distance from
A, B, and C?

5. Can a circle be drawn which contains each of the vertices of this triangle?
If so, draw it.

Next draw an obtuse scalene triangle and repeat the process for locating
point O. The teacher should be certain that in observing the groups and
their work that they have recognized that O is the point of concurrency
for the three perpendicular bisectors, AC, BC and AB. The students should
be asked to discuss the following in their groups.

1. Why does the crease made produce the perpendicular bisector of each of
the sides in ABC?

2. Where would the point O be in a right triangle?

3. If ABC is scalene, none of the perpendicular bisectors passes through
a vertex. Would this be true of an isosceles triangle? an equilateral triangle?

Redirect the students attention to the beginning question. Are you able
to draw a circle through the vertices of any triangle?

The teacher might then follow up with a GSP lab having the students create
many triangles to test their conjectures.

CONSTRUCTION:

As shown below draw a triangle ABC and construct the perpendicular bisectors
using GSP.

GSP LAB: Click here for a GSP sketch to explore
the following questions.

Let F be the intersection of the two perpendicular bisectors. Trace it.

Observe the trace of point F as you drag point C. What do you conjecture
the trace of point F is?

Measure the distance from F to other points. What relationships do you find
among the distances?

Construct the object which you conjecture is the trace of point F. Check
to see if F travels along the object as you move point C.

Construct the circle with center F through point C.

What do you observe? How is this exploration related to earlier investigations?

Conclusions: The circumcircle of a triangle is the circle passing through
all three of the vertices of the triangle. The center of the circle is the
circumcenter of the triangle.

Classroom followup: Prove that the perpendicular bisectors of the sides
of a triangle are concurrent in a point this is equidistant from the three
vertices of the triangle.

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