Tangent Circles

By Taylor Adams

 

Given two circles, C1 and C2, letís find a circle tangent to both.

 

First, construct a circle with its center on C2 with the same radius as C1.

 

Next, construct a line through the center of C2 and the center of the new circle, point C.The intersection of this line with the new circle outside of circle C2 is labeled point D>This is can be thought of as a line through the diameter of C2.

 

Next, construct a line segment from the center of C1, point A, and point D.

 

Next, construct the perpendicular bisector of line segment AD.Label the intersection of the perpendicular bisector with the diameter of C2 as point E.This point will be the center of the circle tangent to both circles C1 and C2.

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Finally, construct a circle with center E with radius EC.This is the circle that is tangent to both circles C1 and C2.

 

Letís look at the loci of the centers of the tangent circle.

To do this, we can animate the circles in GSP and trace the tangent circleís center to form the loci of the centers.

 

When one circle is inside of the other, we can see that the loci of the center is an oval shape.

We want to prove that the loci is an ellipse.To do this, we want to show that the distance from a fixed point to the center of the tangent circle to another fixed point is constant.We will use the centers of the two original circles as the fixed points.

This gsp file shows that this distance is constant.

 

When the two circles cross each other, we can see that the loci of the center of the tangent circle is an oval shape.

We want to show that it actually is an ellipse.To do this, we want to show that the distance from a fixed point to the center of the tangent circle to another fixed point is constant.We will use the centers of the two original circles as the fixed points.

This gsp file shows that this distance is constant.

 

When the two circles are disjoint, the loci of the center of the tangent circle creates a hyperbola.

To prove that this acutally is a hyperbola, we want to show that the difference of the distance from a fixed point to the center of the tangent circle and the distance from a second fixed point to the center of the tangent circle is constant.We will use the centers of the two original circles as the fixed points.

This gsp file shows that the difference of these distances is constant.

 

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