Tangent

A line that contacts an arc or circle at only one point.

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 "tan-gen-shull").

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:


Two external tangents See Constructing tangents to two circles (external) with compass and straightedge.	Two internal 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

See Constructing tangents through an external point for demonstration of how to draw the two possible tangents to a circle through an external point, using only a compass and straightedge.
See Constructing tangents through a point on the circle for demonstration of how to draw the tangent through a point on the circle , using only a compass and straightedge.
See Constructing tangents to two circles (external) for demonstration of how to draw the two possible tangents to two given circles, using only a compass and straightedge.

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.

Tangent

Tangents to two circles

Construction

Other definitions

Other circle topics

General

Equations of a circle

Angles in a circle

Arcs