by Donny Thurston
For this write-up, we are going to explore the process of constructing circles that are tangent to another circle and an arbitrary point on a line. As we construct the figures, and determine why we have succeeded in creating the contstruction, we will look to observe interesting behaviors and patterns within our figure. As we make conjectures about what we see, we will refer to our construction to understand if we are correct.
In this construction we will see that we can actually create two circles that fulfill our conditions.
So, lets begin by seeing our situation.
Here we have our circle with center, K, and our line with an arbitrary point, P. By observing the figure, we may be able to imagine the two circles that we are going to be able to create. One will be small, between the circle and the line. The other will be larger, and will include our circle, being tangent to the circle on the "other side" of it, in a manner of speaking. The question is, how do we find the center of either tangent circle, given that we don't know at which point it intersects our circle.
Let us start with what we do know. We know that the centers of both of our target circles must be on the line perpendicular to our given line, as it it goes through point P.
Something that can help us conceptualize this is to imagine what we can find. It would not be hard to find the circle that includes both P and K, as we could connect these two points with a segment, and then the perpendicular bisector of that segment would intersect our perpendicular line at P at the center of the circle.
This is, of course, not what we want, but it can help us get there. Instead of intersecting at K, we want to intersect as a single point on the circle, exactly one radius away from K (in either direction, for each circle).
So, lets "overshoot" P, in order to compensate! By connecting K to a point exactly one radius past P, we would make a circle that is too large, going through K, as well as going past P. However, by choosing to shorten the radius of our new circle to end up at P (instead of the point past it) the radius fails to reach K as well, instead ending up exactly one radius short. This of course, is right on the circle!
Here we see the incorrect circle in red-dashed, and the correct circle, our goal, in purple.
Similarly, we can find the larger circle by "undershooting" P by exactly one radius. The accompanying compensation, similar to our last construction, will take a circle that is too small (and goes through K). When the radius is taken out to P, it will also end up on the approximate opposite side of the circle, resulting in the larger goal circle that we were looking for.
So, we did it. Here are what they both look like one the same image, the first figure has our helping constructions, in dotted lines, and our second just has our final products.
So, what can we do with this? Are there any interesting relationships or actions occurring? When we perform this same construction (really done the same way!) with two circles (as opposed to a circle and a line), we see some interesting patterns concerning the center of our new circle, when the point of tangency is moved along one of our circles. See more information about how to make that construction here.
So what about when we have a circle and a line, and have created a circle that is tangent to both? How is this different from when we have two circles?
To follow along, feel free to use the GSP sketch here. Here we have both constructed circles, along with several important constructions that we used when making them.
Well, lets observe what happens when we trace the center of one of our tangent circles, and move point P along our line. The red line will mark the path that the center of our circle takes.
That looks an awful lot like a parabola. So here, is a conjecture: that point, the center of the circle that is tangent to both a circle and a line at a point P, follows the path of a parabola as P travels along the line.
How can we prove this? What do we need to have a parabola? Specifically, we will be looking for a focus and a directrix. Recall that a parabola is defined as the set of points that are equidistant from a focal point and a directrix, a line.
So here is my next conjecture: The focus is the center of the circle, K, and the directrix is the line that P rests on. Let us take a look at a figure to see if we can find a counterexample.
For this figure, I included both circles, and the paths traced by each center. We notice 2 problems with my previous conjecture. First, I did not identify for which parabolic shape I was referring to when identifying my focus and directrix. They may both have the same focus, but they both cannot have the same focus and directrix, because they are distinct. Second, it appears that neither point is mid-way between the proposed focal point and the line that contains P.
However, I am still convinced that both of those figures are parabolas. Let us go back to our construction process, detailed above, for assistance. Let us look at just one circle, for simplicity.
Let us consider how we constructed the center of the constructed circle, or the point in question. This point is not equidistant from K and P, because we had "overshot" P, as discussed earlier. Instead, this point is equidistant from K, and the point on the opposite side of the circle with center P. The distance from the point in question to K is the radius of our circle (the distance from the point and P), plus the radius of the given circle. Therefore, this point is always equidistant from K and the point one radius past P. This is shown by the red arrows.
As P moves along the given line, the point one radius past P (perpendicular from the given line) makes a line parallel to the given line (seen above in orange). The point in question is therefore equidistant from the point K and this line, as per our construction. This means that this point, the center of the circle tangent to a given circle and a given line, does indeed follow a parabolic path, with focus K, and with a directrix (in orange) that is a line parallel to and one radius from the other side of the given line! We were right, we do have a parabola.
So what about our other circle?
We are able to see from our construction, and a very similar argument as for our first circle, that the path the center of this circle follows must also be a parabola. In this instance, our construction required us to create a point that equidistant from K, and one radius away from P, "above" our given line. By a similar argument as with our previous circle, this point also creates a line (in orange, again) when P is moved along our given line. Therefore, the center of our circle must be equidistant from point K and that line, and therefore takes a parabolic path as P is moved along the given line. As a result, K is the focus, and the orange line shown, one radius closer to the center of the circle than P, is our directrix.
So, in constructing circles that are tangent to a circle with center, K, and any a given line at point P, we see some interesting facts. The centers of each of these circle follow a parabolic arc as P moves along the line. This is similar behavior to what would happen if we created a circle that is tangent to two disjointed circles instead of a circle and a line. By examining this, we can consider this situation as a limiting case, as if we had two circles, but one of infinite radius.
This final figure contains everything discussed here, the circles, some basic marks included in the construction and the directrix for each parabola.
You can find the accompanying GSP sketch here.
