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