Join Circles by Given Segment

Two circles C(O) and C(O'), with centers O and O', and a line L are given. Locate a line parallel to L, so that the distance between the points at which this line intersects C(O) and C(O') is given to a given value a.

In the applet below, the line and distance data are incorporated into a vector parallel to the line and having the given length. That vector can be dragged as a whole parallel to the previous position or rotated around any of the endpoints by dragging the other endpoint. The two circles can be dragged as a whole, or have their radii changed by dragging their centers.

The problem is a classic one and is among the simplest that illustrate the applications of the Translation Transform.

The problem may have 4, 3, 2, 1, or zero solutions. This is how they are found.

Form a vector v in the direction of line L and having length a. There are two,zero,one,two such vectors pointing in opposite directions. Translate circle C(O) into C(O'') by vector v. Circles C(O') and C(O'') may have 2, 1, or no common points. In case there are 2, let's denote them P' and Q'. These are the images under the translation by vector v of points P and Q on C(O), so that both PP' and QQ' are parallel,equal,perpendicular,parallel,adjacent to L, each measuring a in length.

References

1. I. M. Yaglom, Geometric Transformations I, MAA, 1962, Problem 1

Copyright © 1996-2010 Alexander Bogomolny