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 |AP| = r * |BP| 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.

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.