# Secant

A secant is a line that intersects a curve at a minimum of two different points.

The secant line above cuts (intersects) the curve at three distinct points.

## Secants and circles

In Geometry, secant lines are often used in the context of circles. The secant line below, in red, intersects the circle with center O, twice. If a line intersects a circle at only one point, it is called a tangent line. The blue line is a tangent line for the circle.

Two nonparallel secant lines for a circle can intersect in one of three ways:

Knowing secant lines and angles formed by their intersects is beneficial when placing T.V. cameras in positions that best capture a television news cast.

## Intersecting Secants Theorem

Some interesting properties exist when two secant lines intersect outside of a circle.

For two secant lines that intersect outside of a circle, the product of the measures of one secant line segment and its external segment is equal to the product of the other secant line segment and its external segment:

In the figure above, AB and AC are secants for circle O that intersect at A. AD and AE are external segments. The Intersecting Secants Theorem states AB·AD = AC·AE.

Additionally, there is a relationship between the angle created by the secant line segments and the two arcs, shown in red and blue below, that subtend the angle.

Secants AB and AC form ∠BAC which intersects circle O, creating arcs BC (in red) and DE (in blue). The relationship between the angle and the arcs is .