A line that is tangent to two given circles.
There are five possible situations :
(1) If the two circles are disjoint circles, then there are 2 external tangents (\(d_{1}\) and \(d_{2}\)) and 2 internal tangents (\(d_{3}\) and \(d_{4}\)) :
(2) If the two circles are externally tangent, then there are 2 external tangents (\(d_{1}\) and \(d_{2}\)) and 1 internal tangent (\(d_{3}\)) :
(3) If the two circle intersect, then there are 2 external tangents (\(d_{1}\) and \(d_{2}\)) :
(4) If the two circles are internally tangent, then there is 1 external tangent (\(d_{1}\)) :
(5) If one of the circles is inside the other circle, then there is no tangent to the two circles :
Example
The line ABabove is tangent to the two circles \(O_{1}\) and \(O_{2}\) at points A and B, respectively.