Assignment 7

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.

I chose to write up problem number four from this assignment:: Discuss the loci of the centers of the tangent circles for all case you construct.

Given two circles and a point P on the first circle, there are actually two different circles tangent to the first circle at P and tangent to the second circle:

For two circles to be tangent at a point P, their tangent lines at P must conicide. From basic Euclidean geometry, we know that the tangent line at the point P of the circle centered at C1 is perpendicular to the segment PC1. Thus any other circle tangent to C1 at P must be centered on the line through P and C (colored red in the folowing diagram):

For this circle to also be tangent to the circle centered at C2, it must also be centered at a point Q on the red line where QP is congruent to QC2.

Circle 1

If we translate the circle centered at C2 to be centered at P, then it intersects the red line at a point R outside the circle centered at C1. Any point on the perpendicular bisector of RC2 (shown in red) is the same distance from C2 as it is from R. At the intersection Q of the two red lines, since the circle centered at C2 and the circle centered at P have the same radius, QS is congruent to QP, and since C2, S, and Q all are on the same line, the circle centered at Q and the circle centered at C2 are both tangent to this circle.

Circle 2

Similarly, if we translate the circle centered at C2 to be centered at P, then it intersects the red line at a point R inside the circle centered at C1. Any point on the perpendicular bisector of RC2 (shown in red) is the same distance from C2 as it is from R. At the intersection Q of the two red lines, since the circle centered at C2 and the circle centered at P have the same radius, QS is congruent to QP, and since C2, S, and Q all are on the same line, the circle centered at Q and the circle centered at C2 are both tangent to this circle.

The locus of Q

Notice that in both cases, when Q is between R and C1, length(RC1) = length(RQ) + length(QC1) = length(QC2) + length(QC1). As P moves around the circle centered at C1, length(RC1) remains constant, which means that length(QC2) + length(QC1) also remains constant. By the definition of ellipse, the locus of Q as P moves around the circle centered at C1 must be an ellipse.

If Q is not between R and C1, then length(RC1) = length(QC1) - length(QC2) or length(RC1) = length(QC2) - length(QC1), in which case the locus of Q is a hyperbola. This GSP file constructs the locus of Q for circle 1 and circle 2.

What happens if Q = C1 or Q = C2?

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