
Dirichlet Regions for Three Points:
The loci of circle intersections trace the boundaries of three Dirichlet regions, each containing one of the points A, B, C. The Dirichlet region containing A, for example, is the set of points P which are closer to A than to any of the other points B or C. It follows that the boundary between two Dirichlet regions is the set where distances to two of the points A, B, C are equal and less than the third distance. Finally, the corner or vertex is the point where the 3 distances to A, B, and C are equal. We recognize this as the circumcenter of triangle ABC. Since the circles have the same radius. The points on the circles have the same distances to the corresponding centers, so the intersection points of two circles are equidistant from two of the three points A, B, C.
Dirichlet Regions for Four Points
Dirichlet Regions for Five Points
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