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.
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.
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.
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.
a. First, a right angle must be constructed through point P with the radius of circle K to create the point of tangency.
b. One tangent circle can easily be constructed using the midpoint of segment KP as its center.
c. The second tangent circle can be found on the other side of line PB.
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.
e. Now the second tangent circle can be constructed using the intersection of line AP with the angle bisector as its center.
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.