Tangent Circles

by Jamila K. Eagles

This exploration begins with the problem: Given two circles and a point on one of the circles, construct a circle tangent to the two circles circles with one point of tangency being the designated point.


In this case, our arbitrary point is E and the construction looks like the picture below. We know that the segment connecting the centers of the two given circles is always the length of the sum of the radius of the desired circle plus the radius of the given circle that did not have the specified point. The same distance can be laid off along the line through the given point from the center of the desired circle through a point on a constructed circle with center at the given point , and radius of the given circle that did not contain the specified point. Now connecting center point of circle C and the center point of circle E, we have an isosceles triangle. The center of our desired circle lies on the perpendicular bisector of the base of our isosceles triangle. Using GSP, we can construct a circle with center at the point where the perpendicular bisector of the base of our triangle and the line that goes through the given circle with the specified point intersect. The radius of our desired tangent circle is equal to the segment formed between the intersection and point E( since the circle must go through point E).

Now we want to consider the locus of the center of our circle tangent to two given circles. Using GSP, we can trace the center point of the new circle as the given point E travels around our given circle A (dark blue). Tracing this point giveus us an ellipse with foci at the centers of the given circles(light blue).

Now we want to look at other cases. We will take a look at what happens when the given circles are two intersecting circles. Using GSP to trace the center of our tangent circle(pink), We find that the locus of our center point is again an ellipse with foci at the centers of the given circles (light blue).

Our final case is the case where we have disjoint circles. The tangent circle constructed is formed in pink in the picture below. By tracing the center of our tangent circle once again, we see that this time the locus of the point forms a hyperbola, but just like the previous cases, the foci are at the centers of the given circles.



If given two circles and a given point on one of the circles, a new circle tangent to both circles through the given point can be constructed. Using geomerter's sketchpad, many concepts can be explored. In my exploration, I found that depending on the case, the locus of the center point of a circle tangent to two given circles can be an ellipse or a hyperbola, both having loci at the center points of the given circles.


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