# Tangent Circle

## By Na Young

Given two circles A and B. Construct a circle tangent to the two circles

with one point of tangency being the designated point.

### First, we make a line connecting a point A and a given point on the circle and then call the intersection point C. We construct a circle C which is the same radius with a circle B. We call the intersection pointof a circle C and a line AC a point E. Next, make a midpoint of a segment BE and a perpendicular line from that point. We can construct a point D in this way. The point D is a center of circle tangent to the two circles.

The locus of a center of a tangent circle is an ellipse.

It can confirm using GSPÕs animation button.

Next, we will make a different tangent circle.

LetÕs construct the tangent circle to two given circles when the smaller circle is internal to the tangent circle.

First, we construct a line through a center and one point of the big

circle. Call the intersection point B and make a circle of a same radius of a small circle A. Then we obtain a point C and construct a parallel line of a segment AC from a point B. The intersection point of a parallel line and a small circle is D. This is a tangent point that we want to find. Connecting D and A, we can find a center of a  tangent circle O.

The locus of a center of a tangent circle is an ellipse ,with GSP, we can animate around the big circle and trace the locus  of the center as follows:

In this investigation I found an interesting method using GSP.

At first, I donÕt know how to think and to use GSP. But seeing a screen of GSP and using a pencil I draw the imaginary picture. In the picture I draw lines and segments and through the experience, I found answers of problems.

Through this problem solving, I knew how to use GSP and how to think using GSP. Before solving this problem, I think GSP is just a showing mathod, however, I became to know an important of the investigation.

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