32b. Construct a Triangle ABC Given the Length of AB, the Ratio of the Other Two Sides and a Line through C
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 .[more]
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 .
The circle is the Apollonius circle of the points and with respect to the ratio .[less]
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 .
 E. J. Borowski and J. M. Borwein, Collins Dictionary of Mathematics, New York: HarperCollins Publishers, 1989 pp. 21–22.