Rayen Antillanca. Assignment 7

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.

 Figure 1 The green circle is tangent to the two purple circles, and it does not surround the inside circle.

 The blue circle is also tangent to the two purple circles but it does surround the inside circle Figure 2

First Investigation

The locus of the tangent circle’s center is an ellipse, as it is shown in red by figures 3 and 4.

 Figure 3 Figure 4 In figure 3 we observe that BC = BD.  Then, we can see that AB + BE = constant since it is the sum of the radii of the given circles.  Thus, the locus of B is in fact an ellipse by definition.   In figure 4, we may elaborate a similar argument to prove that the locus of B is an ellipse. Let L1 be the tangent line through C, and let L2 be the tangent line through F. L1 is perpendicular to the radius of the given circle with center at A, and L2 is perpendicular to the radius of the given circle with radius at E.  Consequently, L1 and L2 are also perpendicular to the radii BC and BF, respectively, so that A, B, and C and B, E, and F are collinear points, respectively.  Finally, consider                     (1)   Since BC and BF are radii, and consequently BC = BF, we obtain that                                                                   (2)   The relation (2) is constant because it is the subtraction between the radii of the given circles. Consequently, the locus of B is an ellipse by definition.

You can explore the locus of figure 3 here and the one in figure 4 here

 To complete the construction the next picture shows the green and blue circles together. To draw this figure you can use this tool. Figure 5

 The loci of the two tangent circles’ centers are two ellipses, for the green circle is the green ellipse and for the blue circle is the blue ellipse. Figure 6 With this animation, you could explore the following situations:   1) What happens when the given circles are tangent to each other, and the smallest one is inside of the largest one?   2) What happens when the given circles are tangent to each other, and the smallest one is outside of the largest one?   3) What happens when the given circles intersect to each other?   4) What happens when the given circles do not intersect, and the smallest one is outside of the largest one?   This animation will permit you to explore this

Second Investigation

 What happens with these two tangent circles (green and blue) when the two given circles (purple) intersect to each other? Figure 7

 The loci the tangent circles’ centers shown in figure 7 Figure 8 As shown in the above figure, the locus of the green tangent circle’s center is an ellipse, while the locus of the blue tangent circle’s center seems to be a “know curve.” It is easy to show that the latter locus is a hyperbola.  Consider the Figure 9 shown below: Figure 9 Let L1 be the tangent line through M, and let L2 be the tangent line through N.  L1 is perpendicular to the radius of the given circle with center at C, and L2 is perpendicular to the radius of the given circle with radius at A.  Consequently, L1 and L2 are also perpendicular to the radii BM and BN, respectively, so that B, C, and M and B, A, and N are collinear points, respectively.  Finally, consider                  (1)   Since BM and BN are radii, and consequently BM = BN, we obtain that                          (2)   The relationship (2) is constant because it represents the subtraction between the radii of the given circles.  Therefore, the trace of the blue tangent circle’s center is a hyperbola by definition. This animation will permit you to explore this.

Third Investigation

 What happens when the two given circles (purple) are completely separate from each other? Here the green circle passes between the two given circles. The blue circle encircles the both given circles. Figure 10

 The loci the tangent circles’ centers shown in figure 9 Figure 11 As the trace of the centers of the green and the blue circles show, the loci of the green circle’s center and the blue circle’s center are both hyperbolas.  This animation will permit you to explore this

 Note: All figures and tools on this web page were made with The Geometer’s Sketchpad