Are there a certain number of arbitrary points, such that there exists a unique circle through them? To explore and conjecture an answer to this question, students will construct circles passing through various numbers of given points. This activity was adapted from an activity developed by Sean Jones, a fourth year pre-service teacher at the University of Virginia in Spring 1998.

Mathematics: Students will investigate the following theorems:

• Infinitely many circles can be constructed through any single point.
• Infinitely many circles can be constructed through any two points.
• Any three points determine a unique circle.

Mathematical Thinking: Students will be asked to hypothesize, conjecture, generalize and prove the construction methods of various circles.

Technology:  Students will learn to use several Sketchpad commands: Point Tool, Segment, Point On Object, Point At Intersection, Point At Midpoint, Perpendicular Line, Circle By Center+Point, Distance, and Length.

The following sketch illustrates the constructions of a circle through two and three given points.