An Introduction to Tangent Circles

By Sharon K. OŐKelley

1.  Consider this problemÉ. Using GeometerŐs Sketchpad, find the two lines tangent to a circle from an external point.

Figure 1

a. To solve this problem, a student must understand the concept of tangent. A line is tangent to a circle when it intersects the circle in one point. At that point, the radius of the circle forms a right angle with the tangent line. If the radius forms a right angle with the tangent line, then the segment OP becomes the hypotenuse of the right triangle.

Figure 2

b. Since an inscribed angle is one-half the measure of the arc it subtends, OP must be 180 degrees since angle OAP is a right angle. Therefore segment OP is the diameter of a second circle that passes through the points of tangency. Using the midpoint of OP as its center, a circle is constructed with a radius of one-half the measure of segment OP. Where this circle intersects the first circle are the points of tangency. These points can now be connected to point P thus forming the tangent lines.

Solution

Figure 3

2. What about finding tangent circles? ConsiderÉ.

Given a line and a circle with center K. Take an arbitrary point P on the circle. Construct two circles tangent to the given circle at P and tangent to the line.

Figure 4

a. First, a right angle must be constructed through point P with the radius of circle K to create the point of tangency.

Figure 5

b. One tangent circle can easily be constructed using the midpoint of segment KP as its center.

Figure 6

c. The second tangent circle can be found on the other side of line PB.

Figure 7

d. Keep in mind that in order for the second circle to be tangent to line AB as well as to Circle K at point P, its center must be equidistant from the sides of angle PBD. Therefore, its center must fall on the angle bisector of angle PBD.

Figure 8

e.  Now the second tangent circle can be constructed using the intersection of line AP with the angle bisector as its center.

Solution:

Figure 9

3. What about circles that are tangent to only circles as seen in Figure 11? How are these created? How are they related to the construction of an ellipse or a hyperbola? For a discussion of the relationship between conics and tangent circles, go here to see a GeometerŐs Sketchpad presentation of the topic.

Figure 10

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