Exploring Tangent Circles

By Princess Browne

 

Given two circles A and B we need to construct a third circle C that is tangent to both circle A and circle B. We know that when two circles are tangent they share a common point. Therefore we will start this investigation by creating circle A and circle B.

 

 

 

 

To construct a circle tangent to Circle A and circle B we will construct a point on circle A or a point on circle B. Then we will construct the radius of circle B and a line through the center of circle A. Next, we will construct a circle C from the radius of circle B. We will create a segment from the center of circle B to the point on circle C. Then we will construct a triangle from the two points on the line and the point from circle A. Next, we will find the midpoint of the segment and create a perpendicular line to help with the construction of circle C. We will find the point, where the perpendicular bisector intersects the line that passes through circle B which is the center of circle C.

 

Next we will look at the loci of tangent circles

 

 

The picture below indicates that when the two circles intersect the trace of the loci form is still an ellipse that shows the circles intersection points.

 

 

The next picture will show that when circle A and Circle B does not intersect the trace of their loci forms a hyperbola

 

 

 

For further illustrations, lick the GSP Files below

Script file

When circles intersect

When circles do not intersect

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