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: $$|AS|\cdot|SC|=|BS|\cdot|SD|$$

The converse is true as well. That is: If for two line segments $\overline{AC}$ and $\overline{BD}$ 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: $$|AS|\cdot|SC| = |BS|\cdot|SD| = r^2-d^2$$ 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 $△BSC$: $$\begin{align} \angle ADS&=\angle BCS\, (\text{inscribed angles over AB})\\ \angle DAS&=\angle CBS\, (\text{inscribed angles over CD})\\ \angle ASD&=\angle BSC\, (\text{opposing angles}) \end{align}$$ This means the triangles $△ASD$ and $△BSC$ are similar and therefore

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

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.