My explorations in this area began several years ago in Geometer’s Sketchpad, but quickly moved to GeoGebra. This article really began as my attempt to verify what I had conjectured while using GeoGebra.
First we will take two circles with any radii and any centers. The circles can be disjoint, overlapping, concentric, or any relation to each other. Just be sure that they are different from each other.
Circle 1 with center and radius r .
Circle 2 with center and radius s.
We will consider a point on circle 1 rotating counterclockwise around its center starting at the point . This starting point is on the circle directly to the right of the center. The position of this rotating point can be denoted .
We will also consider a point on circle 2 rotating counterclockwise around its center starting at the point . This starting point is on the circle directly to the right of the center. The position of this point will be given by .
Please take note that the two points have the same orbital speed, but opposite directions.
We will analyze the path followed by the midpoint of the segment connecting these two points. Based on the midpoint formula (actually just a simple averaging), we get our point of interest:
Now comes a bit of algebraic and trigonometric manipulating.
Separating into two equations and multiplying both sides of each by 2 we get:
Applying the negative angle identities, we obtain:
A bit of factoring gets us:
We can now isolate the trig functions (provided ):
A bit of adjusting yields
We can now take the fractional side of each equations and divide each numerator and denominator by 2.
We can know apply the basic Pythagorean identity
This equation is now in the standard form for an ellipse with vertical axis
The values of the elliptical equation pair up as follows:
(We can take note that this requires that r and s not be equal.)
Also since , we can be sure that the center is the midpoint of the segment with endpoints at the centers of the two circles with which we started. In addition, since , we can be guaranteed that every ellipse created this way must have a horizontal major axis.
Our situation is represented here:
This leaves us with a few leading questions: