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

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 .
Construction
1. Draw the line and a line . On choose points and , such that .
2. Construct a triangle such that . Let be the point where the angle bisector at meets . Let be the point where the angle bisector of the outer angle at meets .
3. Let be the midpoint of . Draw the circle with center and radius .
4. The point is an intersection of and .
Verification
The triangles and have the same Apollonius circle.

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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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