Erik D. Jacobson

Tangent Lines | Home

In this exploration, I use
the animate feature to explore lines tangent to two circles. In the GSP sketch below, the purple
line is tangent to the left circle and the intersection of this line with a
perpendicular drawn to the center of the right circle is traced in red. Whenever the red locus intersects the
right circle, then, the purple line is tangent to both circles. In the first case, one can see that
there are four possible lines of tangency.

The second case occurs when
the two circles intersect at a single point. Here there are three intersections of the red locus with the
circle on the right and hence, three possible lines of tangency.

If the circles intersect
twice, then there are two lines of tangency. Note that the loop of the red locus has come entirely inside
the right hand circle.

If one circle is contained within the other and there is only one point of intersection, they there is one line that is tangent to both circles, namely the line through the point of intersection.

Finally, if one circle is contained entirely within the other, then there can be no lines of tangency. The red locus does not intersect the interior circle at all.