Overview: First, I will discuss two different constructions of how to find two tangent circles to a given line and a given circle. I will then look at the locus of points of the centers of the two tangent circle and look for interesting relationships. Finally, I will use these two constructions to discuss some of the issues involved in teaching mathematics and then demonstrate how these two constructions apply to those issues by considering what we can take away from each construction.
Construction 1: First, we are given a line and a circle with a point P that lies on the circle as shown below.
We need to find the two tangent circles to the line and the given point on the circle. We do this by first constructing a line through the center of the circle perpendicular to the given line and a line through the point P also perpendicular to the given line as seen below.
Next we draw a line through the center of the circle and the given point P. The center of the smaller tangent circle must lie along this line. Why?
If we find the point along line j such that the height from the given line to line j is equal to the distance from point P along line j to this point then we will have a circle that is tangent to the given circle and the given line. Click here and hit the animate button to see an animation. In this animation the center will be at the point when PF = PG.
How do we find this point? Let's draw the portion of line m between the circle and line k and the segment between point P and line k and find the midpoint of each segment. Then draw the line through the two midpoints, and this intersection point is the center of the circle that is tangent to the given circle and given line.
|Now let's construct the circle and see if we can find the second circle.|
The second circle we can find by drawing a perpendicular at the point where line k intersects line n. The radius of the circle is then the distance from this perpendicular to where it intersects line j.
So HI is the radius of the second tangent circle. Let's have a look at why this construction gives us two tangent circles. As of yet, I have yet to figure out why this construction works. I will post it when I figure it out.
Construction 2: Let's look at a slightly different construction of these tangent circles for which I do understand the justification of the construction. The construction begins the same way with a given circle and a point P that lies on this circle and a given line. We first want to draw a line through the center of the circle and the point P as shown below.
|Now we draw the tangent line to the circle at the point P. As shown to the right.|
Now, if we draw the angle bisector of PQR this should intersect line l at the center of the circle tangent to the the given circle and the given line.
I have also drawn in the segment that is perpendicular to line K from the point S. We now have that triangle PQS is congruent to triangle TQS by AAS or SAS. This means that PS is congruent to ST and therefore we know that S is the center of the tangent circle with radius length PS. If we bisect the supplementary angle to PQS we get a second point of intersection along line l and this serves as the center of our second tangent circle.
Locus of points for construction 1: Let's now take a look at the locus of points swept out by the centers of the two circles. Can we make any guesses before we use GSP to animate? We know that the center of the circle is equidistant from the point of tangency on the circle and the point of tangency on the line. Does this help? Let's look at the results of the animation.
It looks like the top curve that is swept out is a parabola. It is less clear what the bottom curve is but let's go back to the definition of a parabola and see if it makes sense that that is what we are getting. We know that the center of each circle is equidistant from the point P of tangency on the circle and the point of tangency with the given line. The definition of a parabola is given a line and a fixed point not on that line a parabola is the set of all points that are equidistant from the point and the line. It looks convincing but there is one problem with the definition--namely that the point P is not a fixed point. In other words P is moving around the circle and the centers of the circles are equidistant from a point that is moving. It appears then that these may not be parabolas.
This, however, is an incorrect interpretation. In fact, in this case our gut intuition is correct. These curves are indeed parabolas. Why? If we take our fixed point to be the center of the given circle and take a line parallel to the given line but below it this point and line work as the fixed point and given line as stated in the definition above.
What happens as we move the line to various different places? Let's try by putting the line in the interior of the circle.
When we have the line inside the circle one parabola opens up and the other opens down. Based on this evidence can you guess what happens when the line is above the circle (I will leave this as an investigation for the reader)? If you want to investigate yourself click here and click on the animation button.
In this construction are there any other interesting geometric relationships? Let's focus on the segment perpendicular segment that connects P to the given line. We know that the midpoint of this segment is equidistant from point P and the point on the given line as P moves around the circumference of the circle. If we take the locus of the midpoints as P moves around the circle can you think of what geometric shape this will trace out?
As we trace this midpoint, we get what I think is an ellipse. To investigate this animation yourself click here. I am not sure what the two focal points would be so I will leave this as an unproved conjecture. What relationship does this ellipse have to the two parabolas? Let's again take a look at the GSP animation to find out.
It is tangent to both of the parabolas which the centers of the circles sweep out.
Locus of points for construction 2: The locus of points for the second construction gives us the two parabolas but it doesn't give us this interesting relationship between the ellipse and the two parabolas. Now let's think about the advantages and disadvantages in the classroom of these two constructions.
Looking at these two constructions from a teacher standpoint: The second construction is a bit more clean than the first. It requires less steps and the justification of why it works can be easily seen (hopefully by the teacher and also is easier to explain to students). One of the difficulties of allowing students to arrive at solutions of their own is that they may come up with solutions for which you cannot readily justify but which work. Therefore, it is easy to fall in to a pattern of too much directed guidance when solving problems and not allowing your students to come up with creative solutions to a problem. What is lost when you tell students how to solve a problem rather than give them the skills to problem solve themselves? Not only are students learning only through repition often times interesting mathematics is missed when we look for only one way to arrive at a mathematical solution. This problem provides a perfect example as the second construction doesn't allow us to see the relationship between the ellipse as the two parabolas. In both constructions, the problem is answered. It is valuable to allow students to see if they can come up with a construction with the teachers guidance. Having several different constructions provides the opportunity to discuss the strengths and weaknesses of each. The strength of construction 1 is that it allows us to further investigate mathematical relationships that construction 2 does not allow. The strength of construction 2 is that it is easily justifiable and solves the problem efficiently. Multiple solutions of a problem often bring these issues to the floor and provide for great discussions in class.