The Apollonius Circle of a Triangle
An Apollonius circle is the locus of the apex of a triangle on a given base, the other two sides of which are in a fixed ratio.[more]
In this Demonstration, is the length of the base of the triangle and the ratio , where . The circle is the circumcircle of a triangle , where and are points of intersection of bisectors of the internal and the external angles at ( and ) and the line through . The bisectors form a right angle.
Let and be points on the line through , such that . Then and .
Thus , . The point divides the segment into and . The diameter of is and depends only on and .
Let be the midpoint of . The power of with respect to is . Let be an intersection of the Apollonius circle and the circle with its center at and radius . The circles are orthogonal and is tangent to .[less]
 E. J. Borowski and J. M. Borwein, Collins Dictionary of Mathematics, New York: HarperCollins Publishers, 1989 pp. 211–122.