Common Tangents to Two Circles I
What is this about?
A Mathematical Droodle

(Points O, P, Q, R move horizontally. The diagram can also be dragged as a whole.)

Explanation

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Copyright © 1996-2018 Alexander Bogomolny

Two circles may intersect or lie entirely one within or without the other. If one of the circles lies inside the other, the two have no common tangents. Otherwise, the two circles do have common tangents. The number of the tangents may be 2, 3, or 4 depending as to whether the circles intersect in 2, 1 (i.e. they are tangent to each other), or 0 points.

When the number of tangents is 2, they are both external in the sense that, for each, the points of tangency with the circles lie on the same side of the circles' center line OQ.

When the circles touch each other, the line through the point of tangency perpendicular to their center line is also tangent to the circles.

When the circle are away from each other and do not have common points, there is an additional pair of common tangents that intersect the center line between the centers of the circles. These are known as the internal tangents of the two circles.

(Depending on the relative position of points P and R that determine the size of the circles, the tangents shown may be external or, if the configuration allows, internal to the circles.)

The applet illustrates one possible way of constructing the common tangents to the two circles. The method is based on the construction of a tangent to a circle from a point outside the circle. This uses the fact that the radius-vector to the point of tangency is perpendicular to the tangent. If the circle is centered at O, the point is Q and the point of tangency is U, this means that angle AUQ is right. Thus U can be found at the intersection of the given circle with the circle on OQ as diameter. (This step is common with another common tangent construction.)

Now, for the sake of example, assume we seek the external tangents of two circles. The circles are centered at O and Q and have radii rO and rQ. Suppose rO > rQ. Draw a circle C(O, rO - rQ), with radii rO - rQ and center O. The tangents to the given circles are parallel to the tangents from Q to the auxiliary circle C(O, rO - rQ), which can be found as described above.

When we need to construct the internal tangents, the auxiliary circle has radius rO + rQ; the construction is possible provided the circles lie outside each other.

(Elsewhere this construction is discussed in a framework of comparative assessment of a problem and its generalization. The case of the internal tangents is also illustrated on a separate page.)

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Copyright © 1996-2018 Alexander Bogomolny
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