 Assignment #7

Given two circles and a point A that is on one of the circles, we can construct a circle that is tangent to both circles with one point of tangency being point A. The center of our desired circle will be along the line from the center of our large given circle to point A. Our next step is to create a circle congruent to our smaller given circle with its center at point A. From this point we will use the two small given circles to construct an isosceles triangle that will help us to locate the center of our desired circle.  Let’s connect the center of our smaller given circle (Point X) with the intersection of the congruent circle and the line through the center of our large given circle (Point B) The segment formed will serve as the base of our isosceles triangle.  The center of our desired circle will be at the vertex of the isosceles triangle opposite the base.  That center lies along the perpendicular bisector of the base and is located at the intersection of that perpendicular bisector and the line through the center of the large given circle (Point C). We can now construct our desired circle with its center at point C and containing point A. Let’s look at the isosceles triangle.  Notice how the two legs of the triangle are equal:  Each leg equals the radius of the desired circle plus the radius of the small given circle. Now let’s look at all possible tangent circles (red circles) and the locus of the center of all of those possible circles that are tangent to our two given circles.  By tracing the center of the tangent circle as we move point A, we end up with the following : The locus of the center of all possible tangent circles is an ellipse with the foci at the centers of our given circles.  The perpendicular bisector of our isosceles triangle is triangle is always tangent to the ellipse.  Here is what a trace of that perpendicular bisector looks like: An envelope of lines is produced that are all tangent to and therefore create the shape of the ellipse.

Click here for a GSP Script tool for construction of tangent circles.

Now let’s consider the locus of the center of the desired circles when we change the location of our small given circle.  Up until now we have been looking at the tangent circle when one circle is inside another.  What happens when the two given circles intersect? The locus is still in the shape of an ellipse with the foci at the centers of the two given circles.  The portion of the ellipse that is outside of the large given circle passes inside the small given circle.

Now what if the two given circles are disjoint.  That is, what if the two circles are outside of one another and do not intersect? The locus now takes on a different shape, the shape of a hyperbola with the foci at the centers of the two given circles.