InEuclidean geometry, theintersecting chords theorem, or just thechord theorem, is a statement that describes a relation of the fourline segments created by two intersectingchords within acircle. It states that theproducts of the lengths of the line segments on each chord are equal. It is Proposition 35 of Book 3 ofEuclid'sElements.
More precisely, for two chordsAC andBD intersecting in a pointS the following equation holds:
The converse is true as well. That is: If for two line segmentsAC andBD intersecting inS theequation above holds true, then their four endpointsA,B,C,D lie on a common circle. Or in other words, if thediagonals of aquadrilateralABCD intersect inS and fulfill the equation above, then it is acyclic quadrilateral.
The value of the two products in the chord theorem depends only on the distance of theintersection pointS from the circle's center and is called theabsolute value of thepower ofS; more precisely, it can be stated that:wherer is theradius of the circle, andd is the distance between the center of the circle and the intersection pointS. This property follows directly from applying the chord theorem to a third chord (a diameter) going throughS and the circle's centerM (see drawing).
The theorem can be proven usingsimilar triangles (via theinscribed-angle theorem). Consider the angles of the triangles△ASD and△BSC: This means the triangles△ASD and△BSC are similar and therefore
Next to thetangent-secant theorem and theintersecting secants theorem, the intersecting chords theorem represents one of the three basic cases of a more general theorem about two intersecting lines and a circle - thepower of a point theorem.
