# 32. Construct a Triangle ABC Given the Length of AB, the Ratio of the Other Two Sides and a Line through C

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This Demonstration shows how to construct a triangle given the length of the side , the ratio of the other two sides and a line containing .

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Contributed by: Gerd Baron, Izidor Hafner, Marko Razpet and Nada Razpet (July 2018)

Open content licensed under CC BY-NC-SA

## Details

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 .

Reference

[1] E. J. Borowski and J. M. Borwein, *Collins Dictionary of Mathematics*, New York: HarperCollins Publishers, 1989, pp. 21–22.

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