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No tangent line can be drawn through a point within a circle, since any such line must be a secant line. However, two tangent lines can be drawn to a circle from a point P outside of the circle. The geometrical figure of a circle and both tangent lines likewise has a reflection symmetry about the radial axis joining P to the center point O of ...
The secant lines PQ are the approximations to the tangent line. In calculus, this idea is the geometric definition of the derivative. The tangent line at point P is a secant line of the curve. A tangent line to a curve at a point P may be a secant line to that curve if it intersects the curve in at least one point other than P.
This leads to the definition of the slope of the tangent line to the graph as the limit of the difference quotients for the function f. This limit is the derivative of the function f at x = a, denoted f ′(a). Using derivatives, the equation of the tangent line can be stated as follows: = + ′ ().
The tangent line through a point P on the circle is perpendicular to the diameter passing through P. If P = (x 1, y 1) and the circle has centre (a, b) and radius r, then the tangent line is perpendicular to the line from (a, b) to (x 1, y 1), so it has the form (x 1 − a)x + (y 1 – b)y = c.
The tangent-secant theorem can be proven using similar triangles (see graphic). Like the intersecting chords theorem and the intersecting secants theorem, the tangent-secant theorem represents one of the three basic cases of a more general theorem about two intersecting lines and a circle, namely, the power of point theorem.
The secants ′ ¯, ′ ¯ meet on the radical axis of the given two circles. Moving the lower secant (see diagram) towards the upper one, the red circle becomes a circle, that is tangent to both given circles. The center of the tangent circle is the intercept of the lines ¯, ¯. The secants ′ ¯, ′ ¯ become tangents at the points ,.
A bitangent differs from a secant line in that a secant line may cross the curve at the two points it intersects it. One can also consider bitangents that are not lines; for instance, the symmetry set of a curve is the locus of centers of circles that are tangent to the curve in two points.
A secant plane is a plane containing a nontrivial section of a sphere or an ellipsoid, or such a plane that a sphere is projected onto. Secant planes are similar to tangent planes, which contact the sphere's surface at a point, while secant planes contact the surface along curves.