Plane and solid analytic geometry, by William F. Osgood and William C. Graustein.

CHAPTER V

INTRODUCTORY PROBLEMS IN LOCI. SYMMETRY OF
CURVES
1. Locus Problems.* A point is moving under given conditions; its locus is required. This type of problem the student
studied in Plane Geometry. But he found there no general
method, by means of which he could always determine a locus;
for each problem he had to devise a method, depending on the
particular conditions of the problem.
Analytic Geometry, however, provides a general method for
the determination of loci. Some simple examples of the
method have already been given. Thus, in finding the equation of a circle, we determined the locus of a point whose distance from a fixed point is constant. Again, in deducing the
equation of a line through two points, we found the locus of a
point moving so that the line joining it to a given point has a
given direction.
The method in each of these cases consisted merely in expressing in analytic terms -i.e. in the form of an equation
involving the variable coordinates, x and y, of the moving
point -the given geometric condition under which the point
moved. We proceed to show how this method applies in less
simple cases.
Example 1. The base of a triangle is fixed, and the distance from one end of the base to the mid-point of the opposite
side is given. Find the locus of the vertex.
* The locus problems in this chapter may be supplemented, if it is desired, by ~~ 6-8 of the second chapter on loci, Ch. XIII, in which the loci
of inequalities and the bisectors of the angles between two lines, together
with related subjects, are considered.
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