Tangent
From Latin: tangere "to touch,"
A line that contacts an arc or circle at only one point.
(See also Tangent (tan) function in a right triangle  trigonometry).
Try this Drag the orange dot. The blue line will always remain a tangent to the circle.
The blue line in the figure above is called the "tangent to the circle c".
Another way of saying it is that the blue line is "tangential" to the circle c. (Pronounced "tangenshull").
The line barely touches the circle at a single point. If the line were closer to the center of the circle,
it would cut the circle in two places and would then be called a
secant. In fact, you can think of the tangent as the limit case of a secant.
As the secant line moves away from the center of the circle, the two points where it cuts the circle eventually merge into one and the line is then the tangent to the circle.
As can be seen in the figure above, the tangent line is always at right angles to the radius at the point of contact.
Tangents to two circles
Given two circles, there are lines that are tangents to both of them at the same time.
If the circles are separate (do not intersect), there are four possible common tangents:
If the two circles touch at just one point, there are three possible tangent lines that are common to both:
If the two circles touch at just one point, with one inside the other, there is just one line that is a tangent to both:
If the circles overlap  i.e. intersect at two points, there are two tangents that are common to both:
If the circles lie one inside the other, there are no tangents that are common to both.
A tangent to the inner circle would be a
secant of the outer circle.
Construction
Other definitions

In trigonometry, the tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side.
See Tangent (tan) function in a right triangle  trigonometry.

In calculus, a line is a tangent to a curve if,
at the single point of contact, it has the same slope as the curve.
Other circle topics
General
Equations of a circle
Angles in a circle
Arcs
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