The bisector of an angle in a triangle divides the opposite side in the same ratio as the sides adjacent to the angle. To divide the segment in the ratio , construct a triangle with legs and so that . Choose and with so that and . The foot of angle bisector at divides in the ratio . The locus of such points is the Apollonius circle of the triangle and is independent of . The radius of the Apollonius circle of is
if . The radius depends only on and .
 E. J. Borowski and J. M. Borwein, Collins Dictionary of Mathematics, New York: HarperCollins Publishers, 1989, pp. 21–22.