This write-up is for Assignment 7.

Tangent Circles

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Given two circles and a point on one of the circles it is possible to construct a circle tangent to the two circles with one point of tangency being the designated point.

In the image above, the given circles are in black. Hence, we have been able to construct two different circles that satisfy the constraints of our problem. The red circle and the light blue circle are both tangent to the given circles with one point of tangency being a designated point on our larger black circle.

If we examine our construction below, the red dashed segment is the base of an isosceles triangle. The length of the two congruent sides of this isosceles triangle is equal to the radius of our red tangent circle plus the radius of our smaller given circle. The light blue tangent circle was constructed in a similar way. However, the length of the congruent sides of the isosceles triangle in this construction is equal to the radius of the light blue circle minus the radius of our smaller given circle. Notice that the center of the tangent circle is on the perpendicular bisector of the base of the isosceles triangle mentioned above. The center of the tangent circle is also on the line which passes through the center of the larger given (black) circle and the designated point on the larger given circle.

If we animate the sketch above and trace the centers of our red and light blue tangent circles, we see that the locus of the center of each tangent circle form an ellipse. For a GSP sketch with animation please click here.

In the case of our red tangent circle, our smaller given circle is external to the "red" tangent circle. For a GSP script that will produce this result please click here.

Click here for a GSP sketch with the designated point on the smaller circle.

In the case of our light blue tangent circle, our smaller given circle is internal to the "light blue" tangent circle. For a GSP script that will produce this result please click here.

Click here for a GSP sketch with the designated point on the smaller circle.

By using the three GSP sketches that are linked above, it is possible to explore how the locus of centers of the constructed tangent circle behave.

Please look at the image below. When the designated point on the larger black circle is animated around the larger black circle, the locus (blue image) of the center of red tangent circle forms a hyperbola.

If the two given (black) circles intersect, the locus (blue) of the center of the red tangent circle forms an ellipse as the designated point moves around the smaller (black) circle.

If the smaller (black) given circle is moved inside the larger (black) given circle, an ellipse is formed by the locus (blue) of the center of the red tangent circle as the designated point moves around the smaller (black) circle.

If the two given (black) circles have the same center, the locus (blue) of the center of the red tangent circle forms a circle as the designated point moves around the smaller (black) circle.

In the previous four examples, the path of our designated point has been around the circle external to our red tangent circle. Now, look at the next three examples. Compare and contrast these three examples with the four we have just examined. Feel free to go back to the GSP sketch links above in order to manipulate the given circles.

The locus (blue) of the center of the red tangent circle forms an ellipse when then designated point moves around the larger (black) circle.

In the image below, both given (black) circles are internal to the red tangent circle. The locus (blue) of the center of the red tangent circle forms a hyperbola.

In the next example, the two given (black) circles share the center point. As the designated point moves around the larger (black) circle, the locus (blue) of the center of the red tangent circle forms a circle.

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