Intersecting chords theorem

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.

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

${\displaystyle {\begin{aligned}&|AS|\cdot |SC|=|BS|\cdot |SD|\\=&(r+d)\cdot (r-d)=r^{2}-d^{2}\end{aligned}}}$

${\displaystyle \triangle ASD\sim \triangle BSC}$

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 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:

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

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:

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

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 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.