Tangent Circles

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This investigation begins with the following problem.

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.

But, what are tangent circles?

For any two given circles, a tangent circle is a circle tangent to both given circles.  Below are some possibilities (given circles in black, tangent circle in red):

Many explorations can be done with tangent circles.  But I will focus on the hyperbola and the ellipse. Here is a trace of a hyperbola by moving point A.  Click here for the hyperbola exploration.

A hyperbola is the set of all points in the plane the differences of whose distances from two fixed points is some constant.

The ellipse is formed below (the black trace).  Click here for the ellipse exploration.

An ellipse is the set of all points in the plane such that the sum of the distances from to two fixed points is some constant.  The two fixed points are called the foci (plural for focus).