Type | Theorem |
---|---|

Field | Euclidean geometry |

Statement | The product of the lengths of the line segments on each chord are equal. |

Symbolic statement |

The **intersecting chords theorem** or just the **chord theorem** is a statement in elementary geometry 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* and *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 *ASD* and *BSC* are similar and therefore

Next to the tangent-secant theorem and the intersecting secants theorem the intersecting chord 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.