Our first investigation during the Lab in the Fall of 2014 was an exploration of the circumcenter. Given three points, the circumcenter is another point on the same plane that is equidistant to them. For example, given A, B and C:
the circumcenter is located at the point whose distances from A, B, and C are equal to each other. As we moved A, B, or C, the circumcenter will move as well in order to keep equal distances to the three points. To make things easier let us call the circumcenter “circum.”
Trying out motions of the points A, B, and C and noticing where circum goes is a way of exploring how circum responds to these motions. We created stories, such as one in which A, B, C were named after participants in the lab and circum worked as their angel guardian. You can read this story here, as well as a subsequent response written by Circum.
Since the distance between Circum-to-A and Circum-to-B must be equal to each other it turns out that Circum is forced to be on the line perpendicular to the one connecting A and B, passing through the midpoint. All the points on this line are equidistant to A and B:
In addition, the distance Circum-to-C must also be equal to the ones Circum-to-A and Circum-to-B, therefore C must be on a circle centered on Circum: