Dynamic Geometry Module Lesson 4: Circles and Tangents
A Tangent Line to One Circle
One of your prerequisite homework exercises was to construct a line tangent to a circle C through a point P outside the circle. The key here was to recall that the radial line (in red in the figure below) from the center, O, through the point of contact, X, is perpendicular to the tangent line (in blue).
Let's see how to create a dynamic sketch that should induce a student to discover how to use this fact to come up with the construction of the tangent line. This is another great use of the dynamic features of Sketchpad.
Open a new sketch and perform the following steps:
- draw a circle and a point, P, outside that circle,
- create a point A anywhere in the plane,
- draw the line through A and the center, O, of the circle,
- draw the line perpendicular to this, though P,
- construct the point of intersection, Q, of these two lines.
You have constructed a line, PQ, perpendicular to the radial line AO. Moreover, for this choice of A, Q is the only point that will make PQ perpendicular to the radial line AO. The only problem is that this line PQ is almost certainly not a tangent line! To find the tangent line, we need to choose a different radial line and hence a different perpendicular.
Try dragging point A, to get a new radial line, until Q sits right on the circle. At this point PQ will be the tangent line we are looking for. Unfortunately, this doesn't tell us how to make an accurate construction that will create the correct position of Q.
Now select Q and choose Trace Point from the Display menu. Then drag the point A completely around circle C. What is the locus traced by Q during this motion? What points does it go through? Is OP a diameter of this locus and why?
The answers to these questions should lead to the appropriate construction:
- construct the segment OP,
- construct the midpoint of this segment,
- construct the circle centered at this midpoint and going through O and P,
- find the intersection points of this circle with the original circle,
- construct the lines through P and each of these intersection points,
- select and hide all the objects that are not needed in the final figure.
If you try this construction you will get both the tangent lines to C through P.
A Tangent Line to Two Circles
In this section we want to start with two circles, each one outside the other. We are looking for a line that is tangent to both these circles. If you try to draw the picture, you will see that there are essentially two different ways that this can happen. The line can touch each of the circles, with both of the circles on the same side of the line (the blue line in the figure below), or it can touch each of the circles with the circles on opposite sides of the line the red line in the figure below).
The first case (the blue line) is called an external common tangent for the two circles, while the other case (the red line) is called an internal tangent. In the rest of this lesson, we will only work with external common tangents. You will create similar constructions with internal common tangents in the homework exercises.
First, let us try a few experiments to try to draw the common external tangents to a pair of circles and see if they give us any hint as to how we might find a construction. Begin by opening a new sketch and drawing two circles, C and D, each outside the other. Pick any point P in the plane and draw the two tangents to circle C through P.
Now move P until the two lines are tangent to D as well as to C. Can you see anything about where P must be in relation to the two original circles? Try moving the circles and repeating to see if your idea still seems to hold.
Open a new sketch and draw two circles as before. Construct points X and Y on circle C and then draw the radial segments from the center of C to both X and Y. Draw lines perpendicular to each of these segments at X and Y respectively. By construction these must be tangent to C.
Move X and Y until you have found (at least approximately) the two common external tangents. What can you say about this configuration? Do the common tangents intersect? If so, do they intersect in a point on the line between the centers of the circles? If you knew the point of contact, X, of one common tangent, how could you find the point of contact, Y, of the other? Think of the symmetry of the picture.
If we have an external common tangent, then the two radii, one for each circle, out to the points of contact, must both be perpendicular to this line. This means that they must be parallel.
We could try to find the common tangent, then, by drawing parallel radii and joining their endpoints. Open a new sketch and draw the two circles as before. Draw a radius OX for circle C. Then draw a parallel line through the center O' of D and find its intersection, Y, with D so that you can draw a radius O'Y that is parallel to OX. Then construct the line joining X and Y.
This is probably not the common tangent that we are looking for. We will find that by changing the position of X. Drag X around circle C until you appear to have the common tangent.
We now have three different ways to get a pretty good approximation for the common tangent line. Unfortunately, none of these has yet given us a precise construction. In the final sketch you created, select the line XY and choose Trace Line from the Display menu. Now repeat the process of dragging X around circle C. What do you notice about the images of the line XY? Do they all go through a single point? If they do, then how could you find that point? Once you know how to locate this point, how can you use it to find the common external tangent(s)?
Answers to these questions give you a method for finding the external tangents.
- Draw parallel radii OX for circle C and O'Y for circle D.
- Join X and Y with a line.
- Join the centers O and O' with a line.
- Construct the point of intersection, P, of these two lines.
- Construct the tangent line from P to either C or D using the construction method from earlier in this lesson.
The point of intersection of the two external common tangents is called the external center of similitude of the two circles.
Practice this construction several times until you are sure that you have it mastered. Do not hesitate to contact us for help if you are not sure about any step. It is essential that you understand this. Also, be sure that you hide all auxiliary lines and points. At the end of the construction the only new objects on your sketch should be the to tangent lines, their intersection point and their points of contact with the circles.
At this stage, it would be good practice to create a custom tool for this construction. Open a New Sketch from the File menu and draw two circles as usual. Complete the entire sketch, constructing the common tangents, and hiding the appropriate auxiliary constructions. Once the sketch is complete, do an Edit, Select All command, and then click on the Custom Tool button, selecting Create New Tool. Name the new tool "external center" and click on the box for Show Script View. You will now see a step by step listing of each of the actions you have taken to create the external centers of similitude. Check the beginning of your script to see what the Given requirements are for this script. This list will depend on how you did your construction, but it will probably contain at least the two circles and their centers. As described in the section on custom tools in Module 4, when you want to replicate this construction for any other two circles, you use the Custom Tool button, select the "external center" tool, and ask it to Show Script View. Then in the script view window, click on All Steps. This method will work particularly well as you do Lesson 4, Excercise 2 in the homework, where you need to create three different centers of similitude in the same drawing.
When you want to use the script, you need to select the corresponding objects on a sketch, in the order they appear in the Given list, and the click either PLAY or FAST on the script. Be sure to save your script. Then try it out a few times to make sure that you know how to use it and to make sure that it works correctly. There is no good way to edit scripts since they are just recordings of actions. If you need to make a correction or alteration, you need to start over again with a new recording of the script.
You should notice that as long as one circle is not completely inside the other and the circles do not have the same radius, then the construction works properly. What happens if the circles have the same radius? Is there a common tangent line(s)? What happens if one circle is inside the other?
This theorem has to do with what happens when we have three circles. Open the sample sketch for this lesson (See file ex4_1.gsp). It shows three circles c1, c2, and c3. The red lines are common tangents to c1 and c2, the blue lines the common tangents to c2 and c3, and the green lines the common tangents to c1 and c3. You can move the centers of any of the circles and adjust their radii by moving the points labeled A, B, or C.
Label the center of similitude of c1 and c2 as X, the center of similitude of c2 and c3 as Y and the center of similitude of c1 and c3 as Z. Do you notice any relationship among these three points X, Y, and Z? Move the configuration and see if this still seems to be the case.
The three centers of similitude appear to be collinear (lined up on a single straight line). How would you check that? Move the configuration and see if your check still holds. This is Monge's Theorem:
For three disjoint circles of unequal radii with no one circle contained in any other, the three external centers of similitude for the three pairs of circles are collinear.
Complete the exercises in Lesson 4 homework and submit them to the MTL handin system.