Intersecting chords theorem

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:

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 (a diameter) going through S and the circle's center M (see drawing).

inscribed angles over AB

inscribed angles over CD

{\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}}}

This means the triangles △ASD and △BSC are similar and therefore

{\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.