Assignment 7
Tangent Circles
by Emily Bradley
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.
Given: circle A, circle B, and point of tangency C. 


Construct a circle with center C the same radius as circle B. Construct a line through AC and mark D where the line intersects circle C outside of circle A. 

Connect segment BD. Through midpoint E, construct a line perpendicular to BD. Where this intersects line AC, call the point F. 

Now DF = FB and we can construct the equilateral triangle DFB. 

Since the segments DF and FB are equal, subtracting the small radius from both creates two equal segments. Thus a circle with center F, radius FC is tangent to both given circles. 
We have just constructed one case on tangent circles.
The second case looks like this.
In these cases by animating C and tracing point F, either an ellipse or a hyperbola is formed, given in the chart below.
An ellipse is the locus of all points of the plane whose distances to two fixed points add to the same constant. These two points are on the ellipse's major axis, on either side of the center. Each of these two points is called a focus of the ellipse. These two foci are the centers of the given circles points A and B. The sum of the distances from any point of the ellipse to those two points is constant and equal to the major diameter, i.e. AF+FB is constant.
Hyperbolas are made up of points, the difference of whose distance from two foci is constant, i.e. AFFB is constant. The foci again are the centers of the given circles, points A and B. a hyperbola is a smooth planar curve having two connected components or branches, each a mirror image of the other and resembling two infinite bows.
Case 1  Case 2  

Containing  
Ellipse 1a  Ellipse 2a  
Intersecting  
Ellipse 1b  Hyperbola 2b  
Disjoint  
Hyperbola 3a  Hyperbola 3b 