This write-up is for problems #7 in Assignment 6.

Constructing Common Tangents to Circles

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In this write-up we will attempt to contstruct the common tangents to two given circles.

A line is tangent to a circle if it intersects the circle in just one point.

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

Common intermal tangents intersect the segment joining the centers of the two circles.

Common extermal tangents do not intersect the segment joining the centers of the two circles.

If we begin with two circles that are tangent to each other, we must examine two different cases. One circle could be inside of the other circle. The two circles would be internally tangent. In this case, we will have only one line that is tangent to both circles. The point of tangency will be where the two cirlces and the one common external tangent intersect. The image below is an example of such a case.

The construction of this common external tangent shown above is a relatively simple task. A line is tangent to a circle if the line is perpendicular to the radius drawn to the point of tangency. The centers of the two circles as well as the point of tangency ( of the circles) are colinear. Therefore, by constructing a line perpendicular to the line which contians the two centers and point of tangency (of the circles) at the point of tangency ( of the circles), we have constructed our common external tangent.

Furthermore, the two circles could be externally tangent as shown in the image directly below. In this case there will be one common internal tangent. The point of tangency will be where the two circles and the common intermal tangent intersect.

The construction of the common internal tangent shown above is similar to the common external tangent previously described. Construct a line perpendicular to the segment that joins the center of the two circles at the point of tangency (of the circles). The common internal tangent will be perpendicular to the radius of each of our two circles.

The externally tangent circles will also have two common external tangents as shown below.

The construction of the two common external tangents requires a little more descussion than our first two constructions. It is possible to construct a tangent to a circle from a point outside the circle. This is done by constructing the midpoint of the segment that joins the center of the circle (point A) and our point C outside the circle. Then construct a circle with its center at our midpoint M with a radius congruent to the segment MC. The points of intersection of our original circle A and our constructed circle M will be the points of tangency. We can then construct a tangent line from our point C outside the circle passing through the point of tangency on our circle A. (There will be two such tangent lines. These tangent lines are line CF and line CE). This construction is illistrated below. The red line CF is tangent to circle A and passes through point C.

In constructing the two common external tangents, the construction described above is very helpful if we reduce the size of the radius of both of our given circles by equal amounts so that our smaller circle is reduced to a point. The line constructed througth our point C (center of the smaller original circle) and tangent to our smaller constructed circle (inside the larger original circle) will be parallel to the common external tangent.

For a GSP script that will contruct the common external tangents for two given circles please click here.

The next logical step is to consider two given circles that have four common tangents.

In this case, there are two common external tangents and two common internal tangents.

The construction for the common intermal tangents is similar to that of the construction of the common extermal tangents. In order to construct the common external tangents we reduced the length of the radius of our smaller circle C to the extent that it became a point C. We then constructed a smaller circle with its center at point A inside our larger circle A. The radius of this newly constructed circle has a radius equal to the radius of circle A (the original larger circle) minus the radius of circle C (the original smaller circle). To construct the common internal tangent, construct a new circle with its center a point A and a radius equal to the radius of our original circle A plus the radius of our original circle C. The length of AF is equal to the lenght of AB plus the length of CD.

If we now construct a line passing through point C and tangent to our newly constructed circle (blue dashed circle), it will be parallel to one common internal tangent of our given circles. (black circle A and black circle C).