Tangent–secant theorem

For a circle, the square of a tangent segment’s length equals the product of a secant’s external segment and its entire length.

Power of a point: A signed quantity associated with a point and a circle, equal to the product of the distances along any secant from that point to the circle. The theorem is the tangent case of the general relation between a point and a circle.
Circle: The set of points in a plane at a fixed distance from a center. The theorem applies to a tangent and secant drawn to this curve.
Circle geometry: The study of geometric properties and relationships involving circles. The theorem is a standard tool for solving length problems in circle diagrams.
Intersecting chords theorem: When two chords intersect inside a circle, the products of the two segments of each chord are equal. It resembles the tangent–secant relation but uses an intersection inside the circle.
Intersecting secants theorem: For two secants through an external point, the products of each external segment and its whole secant length are equal. It shares the same power-of-a-point equation, with two secants instead of a tangent.
Radius: A line segment from the center of a circle to a point on its circumference. A radius to the point of tangency is perpendicular to the tangent line.
Coordinate geometry: The use of coordinates and algebraic equations to represent and solve geometric problems. Coordinates can verify the relation by expressing the tangent and secant lengths algebraically.
Tangent–tangent theorem: Tangent segments drawn from the same external point to a circle have equal lengths. It compares two tangent lengths, rather than a tangent square with a secant product.
Similar triangles: Triangles with equal corresponding angles and proportional corresponding side lengths. The tangent–secant relation can be derived by identifying similar triangles formed by a tangent and secant.
Chord: A line segment whose endpoints lie on a circle. The secant’s portion inside the circle is a chord.

Linked from 4 pages

Secant lineBroader topic: It uses secant lines in circle geometry rather than as average-change approximations.