Loci Of Points Satisfying Certain Properties

In geometry, a locus is a set of all points that satisfy a particular condition or a group of conditions - using loci, we can visualize and determine the positions of points that meet specific geometric criteria.

To better understand this concept we can, for example, take a circle – it is the set of points (locus) at a given distance from a point. Similarly, the locus of points equidistant from two lines is a line that sits halfway between them. It’s the same for any ellipse, they’re simply a locus of points where the sum of distances from focuses is constant.

To find any locus, one must find all points that satisfiy the given conditions. The end result will be a set of points with a shape that differs depending on the original criteria.

To apply loci in the process of solving Apollonius’s problem, we focus on the conditions that the center-point of the circle we are seeking must satisfy. By first finding the locus satisfying only a subset of the criteria and then doing the same for a different subset of the criteria we transform the seemingly impossible Apollonius’ problem into a more manageable task of finding a few loci and then taking their intersections as center-points for the solutions to the problem. These final intersections must now be a locus of the starting conditions.

As an example, we will solve for a PCC with two non-intersecting randomly sized circles c and d and point A. To solve the problem, we use the following facts: A hyperbola is a locus where the difference of distances to foci is constant. Furthermore, the solution we are looking for is a circle tangent to all three defined objects, as per the definition of the problem. Now for the solution:

1. We draw a line through the center of the circles and our point A.

2. We mark the midpoint between point A and the intersections of our circles and the newly established line.

3. We construct the hyperbolas which have the point A and centers of the given circles as foci and intersect the new midpoints (K, J). We do this because we need tangency and wish to create a locus of points that have the same difference of distances to points A and C as points K and J.

4. We construct circles which have centers where our loci (hyperbolas) intersect and pass through point A.

Below you can see the methods for finding loci (same distance) for different combinations of two objects: