Tangent Circles

By

Janet M. Shiver

EMAT 6680




Investigation:Given two circles and a point on one of the circles.Construct a circle tangent to the two circles with one point of tangency being the designated point.


Tangent Circles.

Before we begin our construction of tangent circles we must first understand the approach we will be taking.The basic idea behind our construction is that given any two circles, say A and D, and a third circle E that is tangent to both A and D, then the center point of E must lie along the radius of one of the circles (see figure 1).Also note that the length of the segment connecting the centers D and E is the sum of the radii of circle D and circle E (see figure 2).Imagining that this is the length of the leg of an isosceles triangle, we will construct another leg of equal length through points E and B.To do this we must add the length of the radius of circle D to the radius of circle E along segment EB (see figure 3).

We can now see that if we found the midpoint of CD that the line through the midpoint and the center point E of the tangent circle would be the perpendicular bisector of the base of the isosceles triangle.We can clearly see that the center point of the tangent circle can be found at the intersection of this perpendicular line and the line through EC.

Construction of Tangent Circles

Draw two circles and place a point, A, on the large circle.

Construct a line through the center point B of the larger circle, and point A. Also construct a segment representing the radius of the smaller circle.

Construct a circle with center point a and a radius of the same length as the small circle.Label the point of intersect (outside the large circle) of the new circle and line AB as C.

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Now connect the center of the small circle with point C forming segment DC.Find the midpoint of segment DC and construct a perpendicular through this point.

Finally, mark the point of intersection between the two lines.This will become the center point of the tangent circle.Now construct the final circle using the center point E and the radius EB.

Click Here to see an animation of the construction.




Using a similar construction we can find a second circle that is tangent to both of the original circles.

Locus of Points

Now lets examine the loci of the center of the tangent circles for each of our cases.

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Clearly, the locus of points formed by the center of the tangent circle is an ellipse. It appears that the centers of the two original circles are the foci of the ellipse.Looking at our second construction, we see that if D and A are the foci then the sum of the distances between DE and AE must stay consistent.Click Here and animate the construction to see the measurement of segment DE, segment AE and their sum.

After animating the diagram, it was clear that as the length of segments DE and AE varied their sum was always 3.65cm.

After animating the diagram of our second construction, it was clear that as the length of segments GI and HI varied their sum was always 2.03cm.

Overlapping Circles

We also achieve the same results when the two original circles are overlapping.Notice once again that the center points of ours original circles appear to be the foci for the ellipse formed by the locus of points created by the center of the tangent circle.

Disjoint Circles

Now letís examine the case where the original circles are disjoint.As we can see from the construction below the results are quite different from our other constructions.Instead of a parabola, the locus of points generated by the center point of the tangent circle produces a hyperbola. It appears that the center point of the two original circles are the foci of the hyperbola.

To support our conjecture, we will measure the length of segment DE and segment AE and then find the absolute value of their difference.Click Here and animate the construction to see the measurement of segment DE, segment AE and their difference.

After animating the diagram of our construction, it was clear that as the length of segments DE and AE varied the absolute value of their difference was always 4.29cm.

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