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:

$${\displaystyle |AS|\cdot |SC|=|BS|\cdot |SD|}$$

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:

$${\displaystyle |AS|\cdot |SC|=|BS|\cdot |SD|=r^{2}-d^{2))$$

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

$${\displaystyle {\begin{aligned}\angle ADS&=\angle BCS\,({\text{inscribed angles over AB)))\\\angle DAS&=\angle CBS\,({\text{inscribed angles over CD)))\\\angle ASD&=\angle BSC\,({\text{opposing angles)))\end{aligned))}$$

$${\displaystyle {\frac {AS}{SD))={\frac {BS}{SC))\Leftrightarrow |AS|\cdot |SC|=|BS|\cdot |SD|}$$

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.