Tangent to Two Circles
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 ([latex]d_{1}[/latex] and [latex]d_{2}[/latex]) and 2 internal tangents ([latex]d_{3}[/latex] and [latex]d_{4}[/latex]) :
(2) If the two circles are externally tangent, then there are 2 external tangents ([latex]d_{1}[/latex] and [latex]d_{2}[/latex]) and 1 internal tangent ([latex]d_{3}[/latex]) :
(3) If the two circle intersect, then there are 2 external tangents ([latex]d_{1}[/latex] and [latex]d_{2}[/latex]) :
(4) If the two circles are internally tangent, then there is 1 external tangent ([latex]d_{1}[/latex]) :
(5) If one of the circles is inside the other circle, then there is no tangent to the two circles :
(2) If the two circles are externally tangent, then there are 2 external tangents ([latex]d_{1}[/latex] and [latex]d_{2}[/latex]) and 1 internal tangent ([latex]d_{3}[/latex]) :
(3) If the two circle intersect, then there are 2 external tangents ([latex]d_{1}[/latex] and [latex]d_{2}[/latex]) :
(4) If the two circles are internally tangent, then there is 1 external tangent ([latex]d_{1}[/latex]) :
(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 [latex]O_{1}[/latex] and [latex]O_{2}[/latex] at points A and B, respectively.