In Euclidean geometry, the **intersecting chords theorem**, or just the **chord theorem**, is a statement that describes a relation of the four line segments created by two intersecting chords within a circle.
It states that the products of the lengths of the line segments on each chord are equal.
It is Proposition 35 of Book 3 of Euclid's *Elements*.

More precisely, for two chords AC and BD intersecting in a point S the following equation holds:

The converse is true as well, that is if for two line segments AC and BD intersecting in S the equation above holds true, then their four endpoints *A*, *B*, *C*, *D* lie on a common circle. Or in other words, if the diagonals of a quadrilateral ABCD intersect in S and fulfill the equation above, then it is a cyclic quadrilateral.

The value of the two products in the chord theorem depends only on the distance of the intersection point S from the circle's center and is called the *absolute value of the power of S*; more precisely, it can be stated that:

where r is the radius of the circle, and d is the distance between the center of the circle and the intersection point S. This property follows directly from applying the chord theorem to a third chord going through S and the circle's center M (see drawing).

The theorem can be proven using similar triangles (via the inscribed-angle theorem). Consider the angles of the triangles △*ASD* and △*BSC*:

This means the triangles △

Next to the tangent-secant theorem and the intersecting 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 - the power of point theorem.