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 
.
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.
Snapshots
Permanent Citation
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