tangentsecant

The tangent-secant theorem, also known as the tangent-secant form of the power of a point, is a result in circle geometry that relates lengths from an external point to a circle. Consider a circle with center O and an external point P. A tangent from P touches the circle at T, and a secant from P meets the circle at Q and R, with Q closer to P than R. The tangent-secant theorem states that the square of the tangent length equals the product of the secant segment lengths: PT^2 = PQ · PR.

Interpretation and generalization: This statement is a particular case of the power of a point with respect to a circle. The power Pow(P) is a quantity associated with P that is invariant for all lines through P meeting the circle. For a line through P that is tangent, Pow(P) = PT^2. For a line through P that is a secant meeting the circle at two points A and B, Pow(P) = PA · PB. Therefore, PA · PB is equal to PT^2 for any tangent from P, and more generally, for any two secants through P, the products PA · PB and PC · PD are equal to Pow(P).

Proof sketch: A common approach uses the power of a point or similar triangles formed by the tangent and the secant with the circle. The tangent creates a right angle with the radius at the point of tangency, and the corresponding triangle similarity leads to PT^2 = PQ · PR.

Applications and related results: The theorem is a convenient tool for solving length problems involving tangents and secants, and it underpins broader power-of-a-point results, such as PA · PB = PC · PD for any two lines through P meeting the circle. See also Power of a Point and Tangent Line.