Intersecting secants theorem

For two secants drawn from the same external point to a circle, the product of each secant’s whole length and its external segment is equal.

Secant line: A line that intersects a circle at two distinct points. Each line in the theorem passes through the circle twice.
Chord (geometry): A line segment whose endpoints lie on a circle. The portions between each secant’s circle intersections are chords.
Circle equation: An algebraic equation describing the points of a circle, such as (x − h)² + (y − k)² = r². Coordinates can locate secant intersections and supply the lengths used in the theorem.
Tangent-secant theorem: A circle theorem equating a tangent length squared with a secant’s external length times its whole length. One secant is replaced by a tangent, giving a squared tangent length.
Similar triangles: Triangles with equal corresponding angles and proportional corresponding side lengths. The triangles formed by the secants and chords establish the products’ equality.
External point: A point outside a given geometric figure. Both secants begin at a point outside the circle.
Coordinate geometry: The study of geometric figures using coordinates and algebraic equations. It offers a way to calculate secant lengths when a diagram gives coordinates.
Power of a point: A circle invariant equal to the product of the distances from a point to the intersections of any secant through it. This broader invariant unifies the external, internal, and tangent cases.
Inscribed angle theorem: A theorem stating that an inscribed angle measures half its intercepted arc. Equal inscribed angles in the secant diagram supply the key triangle similarity.
Segment length: The distance between the endpoints of a line segment. The theorem multiplies lengths, not directed segment measures.

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Haruki's theoremRelated: Secant products are a standard route to length identities in circle configurations.