Sharon has a Masters of Science in Mathematics
Have you ever heard of a tangent? The word may bring to mind a conversation taking an unexpected turn, but in math it has a very different meaning. A tangent to a circle is defined as a line that passes through exactly one point on a circle, and is perpendicular to a line passing through the center of the circle. A line that is tangent to more than one circle is referred to as a common tangent of both circles.
A common tangent can be further divided into internal and external common tangents; the difference being that internal common tangents pass through the line segment joining the centers of the two circles and external common tangents do not.
There are six different cases we can look at to talk about all the possibilities of common tangents between two circles. Let's take a closer look at these, one at a time, starting with the fewest number of common tangents and working towards the greatest number of common tangents.
In all the cases below, a will be the radius of one circle, b will be the radius of the other circle, and d is the distance between the centers of the circles. The number of common tangents is n.
How is it possible to have no common tangents for two circles? Well, if one circle is completely inside the other, all of the possible tangents for the smaller circle will pass through two points of the larger circle. Since a tangent to the larger circle can only pass through one point of the larger circle, this means there isn't even a single line that is tangent to both circles.
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Looking at the diagram, we can see that b>a and d < b-a.
Construction: Since there are no tangents, there are no tangents to construct.
If we adjust the geometry of the above scenario slightly, such that the smaller circle is able to just touch the larger circle at one point, we can see that there will be one common tangent, which passes through the single point that is common to both circles.
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In this case, b>a and d = b-a. Also, this tangent is an external common tangent because it does not cross the line segment between the center of the circles.
Construction: The tangent touches the circle at exactly the point the two circles touch, and it's perpendicular to each radius.
Continuing our investigation, what happens when we slide the centers of the circles farther apart? Not too far apart just yet, but enough so that the circles intersect in two points.
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For this scenario, a>b and the distance between the centers of the circles is going to have both an upper and lower bound: a-b < d < a + b. The two tangents are both external tangents.
Construction: Lay a straightedge, such as a ruler or protractor, so that it's touching the center of each circle. Move upwards until just one point of the straightedge is touching the smaller circle. Keep that point relatively still as you adjust the other end of straightedge until it touches just one point on the larger circle. Make any final small adjustments that are necessary to make the line tangent to both circles, then draw the line by tracing the straightedge with your drawing implement. Repeat for the other tangent by moving the straightedge down.
Sliding the centers of the circles apart even further, we can get to the point where we again have just one point that is in both circles, but this time the circles are almost separated.
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You can see that a + b = d, and that there are exactly 3 tangents. There are two external tangents and one internal tangent.
Construction: Draw the external tangents as described in the previous section. The internal tangent will pass through the point that includes both circles and will also be perpendicular to both radii.
The next to last case is where the centers of the circles are far enough apart that the circles are completely separated.
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This is probably the most common case, and is defined by the relationship:
d > a + b. There are 2 internal common tangents and two external common tangents.
Construction: Draw the two external tangents as described in previous constructions. For the first internal tangent, place the straightedge so it's touching the bottom of the left circle and move the right side of the straightedge so it's above the center of the right circle. Keep moving the right side up until there is just one point of the circle touching the straightedge. Make final small adjustments so each circle is touching the straightedge in just one spot. When you have it positioned correctly, trace the straightedge with your pen, pencil or marker.
The last case is a special one, where the distance between the centers of the circle is zero, and the radii of the two circles are equal to each other. In this case, the two circles are actually the same circle and every tangent of the first circle is also a tangent of the second circle.
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For this last case a = b, d = 0. There are an infinite number of common tangents, and they are all external.
Construction: There are literally too many common tangents to draw! To draw a sample tangent, simply pick any point on the circle(s) and draw a line through that point that is perpendicular to the radius of the circle.
A tangent to a circle passes through one point of the circle and is perpendicular to the line connecting that point to the center of the circle. A common tangent is a line that is tangent to two circles. If the tangent crosses the line segment joining the centers of the two circles, it is said to be an internal common tangent; otherwise, it is an external common tangent. The six different cases are summarized in the chart below.
| n | restrictions | internal common tangents | external common tangents |
|---|---|---|---|
| 0 | b > a and d < b-a | 0 | 0 |
| 1 | b > a and d = b-a | 0 | 1 |
| 2 | a - b < d < a + b | 0 | 2 |
| 3 | a + b = d | 1 | 2 |
| 4 | d > a + b | 2 | 2 |
| infinite | a = b, d = 0 | 0 | infinite |
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