The Apollonius Circle of a Triangle

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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.

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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 .

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


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Reference

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



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