Creating a visual interpretation using a plane introduces a number of topics that come to us from geometry. One basic geometric concept is that of the midpoint of a segment. Where geometry uses a compass and straightedge to locate the midpoint, when we have numeric coordinates for two points, the midpoint of the connecting segment is obtained by averaging the first coordinates to get the first coordinate of the midpoint and doing the same for the second coordinate.
Another basic concept is that of distance. Measuring the distance between two points horizontally or vertically is simple, since we can use the appropriate axis as a ruler - the distance is simply the absolute value of the difference between the non-identical coordinates. When two points line on an oblique line, we can calculate their distance with the help of the Pythagorean Theorem. Measuring the horizontal and vertical distances between the points provides us with the lengths of the legs of a right triangle whose hypotenuse is the distance between the two points.
We can extend the idea of distance to write an equation for all the points that are the same distance from a given point. Geometrically, this is a circle. To produce the equation, start with the Distance Formula and replace one of the points with the center of the circle and the distance with the circle's radius. Eliminate the radical from the equation by squaring both sides and the resulting equation describes the circle.