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Tangent Circles

(I’ll try to not go off on a tangent…)

 

By: Carly Cantrell

 

 

 

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.

 

The two circle can either be one inside of the other, overlapping, or completely disjoint.

 

Use my script tool to see for yourself!

 

If you were constructing on your own you would want to do the following:

 

Create any two circles. On one of the circles place a point and make a line through the center of the circle and this point. Next, select the radius of the second circle and the point you just created. Then “Constructą Circle by Center + Radius” and now you will make a point where this circle and the line you constructed intersect outside of the circle. Next, make a segment from the center of the second circle to the point you just constructed. Find the midpoint of that segment. Then create the perpendicular bisector of that segment. Make a point of the intersection of the perpendicular bisector and the first line you constructed. This point is the center of your circle of tangency! Create a circle from that point to the very first point you constructed on the first circle.

 

 

 


 

 

Let’s compare the tangent circle when the two circles are inside one another, overlapping, and disjoint:

 

 

 

 


Let’s investigate the loci of the centers of the tangent circles for all three cases:

 

When the smaller circle is within the other you get the following loci:

 

 

The locus in this case is an ellipse.

We know this is an ellipse because the centers of the two green circles are the foci of an ellipse.

 


 

When the circles are overlapping you get the following loci:

 

 

The locus in this case is an ellipse.

Again, this is an ellipse because the centers of the two green circles are the focal points of the locus ellipse.

 

 


When the circles are disjoint you get the following loci:

 

 

The locus in this case is a hyperbola.

This case is different because when the circles are disjoint the centers of the two circles each act as the foci of a hyperbola.

 

 

 

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