32b. 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 the construction of a triangle given the length of the base , the ratio of the other two sides and a line containing .

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Construction

1. Draw a line and a line through and .

2. Draw the point such that . Draw the points and so that and is parallel to . Let be the point where and intersect. Let be the point where meets .

3. Let be the midpoint of . Draw the circle with center and radius .

4. The point is an intersection of and .

Verification

The circle is the Apollonius circle of the points and with respect to the ratio .

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


Snapshots


Details

An Apollonius circle is the circle defined by the locus of points for which the ratio of the distances from two given points is a fixed number . In this case, the fixed points are and , and .

The radius of the Apollonius circle is

if . This 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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