Circles of Apollonius
Everyone knows that a circle is the set of all points at a fixed distance from a given point, called its center. But the Greek geometer Apollonius of Perga (ca. 262 BC – ca. 190 BC) discovered another way to define a circle. If A and B are two points in the plane, and r is any positive real number (except 1), then the locus of points P satisfying is a circle.
This GeoGebra applet illustrates this theorem of Apollonius. You may drag the points A, B, and P, or use the slider to change the value of r.