Assignment 6: Common Tangents

By Nikhat Parveen, UGA.

Constructing the common tangents to two given circles.

 

First, some basics........

A tangent to a circle is a line, coplanar with circle, that intersects the circle in exactly one point. The point is called the point of tangency. In the figure below, point P is the point of tangency.

A common tangent is a line that is tangent to each of two coplanar circles.

A common tangent can be tangent either internally or externally. A common internal tangent is a common tangent that intersects the segment that joins the center of the two circles. A common external tangent is a common tangent that does not intersect the segment that joins the centers of the two circles. In the figure below, L1 is the common internal tangent while L2 is a common external tangent.

fig.1

Two circles may also be tangent to each other.

Two coplanar circles are tangent to each other if they are tangent to the same line at the same point.

Tangent circles may be tangent either internally or externally. Tangent circles are tangent internally if one lies in the interior of the other (except for the point of tangency). Tangent circles are tangent externally if each lies in the exterior of the other (except for the point of tangency). The figure below shows tangent circles with external and internal tangents.

 

    

       Externally tangent circles              Internally tangent circles

 

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Now back to our goal: How to construct common tangents to two circles with different radii.?

To attain our goal we will start with two circles of different radii, each one outside the other. We are looking for a line that is tangent to both of these circles. If we try to draw the picture you will see that there are essentially two different ways this can happen. As we have seen above in fig.1 the line can touch each of the circles, with both of the circles on the same side of the line (the red line in  fig.1or it can touch each of the circles with the circles on opposite sides of the line the blue line in the figure above).

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. Let's begin by drawing two circles, C1 and C2, 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 C1 as well as to C2. 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. 

Construct points P and Q on circle C1 and then draw the radial segments from the center of C1 to both P and Q. Draw lines perpendicular to each of these segments at P and Q respectively. By construction these must be tangent to C1.

 

Click here for the GSP file

Move P and Q 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, P, of one common tangent, how could you find the point of contact, Q, 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. 

Draw a radius OA for circle C1. Then draw a parallel line through the center O' of C2 and find its intersection, B, with C2 so that you can draw a radius O'B that is parallel to OA. Then construct the line joining A and B.

 

 

O oh! this is probably not the common tangent that we are looking for. We will find that by changing the position of A. Drag A around circle C1 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 we created, select the line AB and trace it.

Now repeat the process of dragging A around circle C1. What do you notice about the images of the line AB? Do they all go through a single point? If they do, then how could we find that point? Once we know how to locate this point, how can we use it to find the common external tangent(s)?

 

Click here for the GSP version

 

By answering the above questions we can find a method for finding the external tangents. We can do so in following steps.

 

Circles with two common tangents

Click here for the GSP construction file.

The point of intersection of the two external common tangents is called the external center of similitude of the two circles

 

Circles with four common tangents

 

The red lines are external common tangents and the blue lines are internal common tangents  to the two given circles

Click here for the GSP construction

The radii OA and OB are parallel to the radii O'D and O'C of smaller circle respectively.

 

Note that common internal tangents cross in between the circles where as common external tangents are exterior to the circles.

 

Three common tangent to two circles

 

Click HERE for GSP construction file.

This figure shows two circles that are externally tangent (each of the tangent circles lies outside the other) at their point of contact.

 

Common tangent to two circles at the point of contact internally

Click here for the GSP construction file

Two circles are internally tangent if one of the tangent circles lies inside the other.

 

Making a script tool.

 

Now we have completed the construction of different common tangents to the two given circles we can make it as a script tool by using GSP tool building features.

Common tangent script tool

 

Let's explore to see if we need different script tools for the different cases.

 

The number of common tangents depends on the relation between the distance and radii of two circles. Given two circles of radii a, b (a > b), with d as the distance between their centers, the following figures show the relation of them to the common tangents.

 

Circles with radii a, b ( a > b) with d as the distance between their centers.

 

When d > (a + b)......

we have four common tangents, two external and two internal

 

When d = a +b .......

we have three common tangents, two external and one at the external point of contact of two given circles.

 

When d = a-b.....

 

 

When d < (a + b)......

 

When d < (a - b)......

no tangents!! to the circles and in fact you can see that one circle is inside the other.

 

Now, do we still need a different script tool for different cases?? Figure it!!

Click HERE to find out.

 

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