Circle Tangents

by

Hyeshin Choi

First, I will construct a GSP tool. First , construct two circles and one is inside the other in this case.

Construct a line pass through center of outer circle.

Name the intersection of outer circle and the line passing through its center as E. Then construct a radius of inner circle and name outside intersection F.

Construct a line between B and F, Then construct a midpoint pf line BF. After that, construct a perpendicular line to BF. When the line intersects on line passing through center of outer center, label it G. Finally construct a circle centerd at G.

Click here for a GSP tool.

Now I am going to explore how the loci work three situations.

1. First I will look at the loci of the center of the tangent circle in different situations. First we’ll look at the case when one circle is inside the otherWhat shape does the locus make? Click here to see animation. Locus of the center D is an ellipse, and the line is the tangent line of the locus.

2. Now let's look at the case when the two circles intersect. What shape does the locus make? Click here to see animation.

Locus of the center D is an ellipse, and the line is the tangent line of the locus.

3. Finally,I will look at the case when the two circles are disjoint. What shape does the locus make? Click here to see animation.

Locus of the center D is a hyperbola, and the line is the tangent line of the locus.