The Apollonius Circle of a Triangle

Initializing live version
Download to Desktop

Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram Player or other Wolfram Language products.

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.


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 .


Contributed by: Gerd Baron, Izidor Hafner, Marko Razpet and Nada Razpet (May 2018)
Open content licensed under CC BY-NC-SA




[1] E. J. Borowski and J. M. Borwein, Collins Dictionary of Mathematics, New York: HarperCollins Publishers, 1989 pp. 211–122.

Feedback (field required)
Email (field required) Name
Occupation Organization
Note: Your message & contact information may be shared with the author of any specific Demonstration for which you give feedback.