In this assignment we will see how to draw a circle through a collection of points. We will also see that three points determine a circle.
To help us, let us recall exactly what is a circle. A circle is a collection of points all of which are equidistant (same distance) from a common point, the center of the circle. The distance between the center and any point on the circle is the radius of the circle.
Problem 1. Draw two points and then draw a circle through the two points. It is not obvious how to do this. Here are some things to think about to help you figure out how to draw such a circle. The whole problem is to find the center of the circle. The center will be equidistant from the two points. Therefore, you need to find a point equidistant from the two points. If you place the center there, you can then draw a circle through the two points. Once you have drawn one circle, figure out how to draw three more circles through the same two points. Look carefully at the centers you found and find a geometric relationship between the points. (The possible centers consist of those points that are equidistant from the two points.)
Describe how you drew a circle through two given points. Make sure to say how you found the center of your circle. Also describe the geometeric relationship between all the points equidistant from two points that you found.
Problem 2. Draw three points and then draw a circle through the three points. In order to do this you need to find the center. To find the center you need to find a point that is equidistant from all three points. To help you do this, label your points numbers 1, 2 and 3. Draw the points equidistant from points 1 and 2, and then draw the points equidistant from points 2 and 3. The center will be common to these two collections of points.
Describe how you drew a circle through three given points, and illustrate your method with two examples. Again, make sure to say how you found the center of your circle. Using your procedure, say why you get only one circle that passes through the points (unlike what happened with two points). This says that three points determines a circle.
Problem 3. Is it possible to draw three points for which there is no circle passing through them? Describe how to draw three such points. Explain why your method fails in this case.