Intersecting chords theorem

The intersecting chords theorem states that when two chords meet inside a circle, the products of the two segment lengths on each chord are equal.

Power of a point: A signed quantity relating a point’s distance from a circle to the products of secant segments through it. The theorem is the interior-point case of this broader circle relation.
Chord (geometry): A line segment whose endpoints lie on a circle. The theorem compares the two parts of each of its intersecting chords.
Circle geometry problems: Geometric problems involving circles, their chords, arcs, tangents, and secants. The theorem converts a circle diagram into a solvable length equation.
Intersecting secants theorem: A circle theorem relating the external and whole lengths of two secants drawn from one exterior point. Unlike intersecting chords, its secants meet outside the circle.
Similar triangles: Triangles with equal corresponding angles and proportional corresponding side lengths. The intersecting chords proof identifies a pair of similar triangles.
Secant line: A line that intersects a circle at two distinct points. Each chord lies on a secant line passing through the interior intersection.
Secant-secant theorem: A circle theorem equating products of external and whole secant lengths from a common point. It extends the same product principle to secants meeting outside the circle.
Ptolemy's theorem: A cyclic quadrilateral’s diagonal-length products sum to the product of its opposite-side lengths. It relates lengths in a circle through a sum, rather than equal chord-segment products.
Inscribed angle theorem: An inscribed angle measures half the angle subtended by its intercepted arc. Equal inscribed angles in the proof follow from sharing the same intercepted arcs.
Segment multiplication: Multiplication of lengths of line segments, often used in geometric proportionality relations. The theorem equates two products formed from four chord segments.

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