Intersecting chords theorem

{{short description|Geometry theorem relating the line segments created by intersecting chords in a circle}}

File:Chord theorem.svg

[[File:Chord theorem power.svg|thumb|upright=1.0|

$\backslash begin\{align\}$

& |AS|\cdot|SC| = |BS|\cdot|SD| \\

={}& (r+d)\cdot(r-d) = r^2-d^2

\end{align}]]

File:Chord theorem proof.svg

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 {{mvar|AC}} and {{mvar|BD}} intersecting in a point {{mvar|S}} the following equation holds:

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

The converse is true as well. That is: If for two line segments {{mvar|{{overline|AC}}}} and {{mvar|{{overline|BD}}}} intersecting in {{mvar|S}} the equation above holds true, then their four endpoints {{math|A, B, C, D}} lie on a common circle. Or in other words, if the diagonals of a quadrilateral {{mvar|ABCD}} intersect in {{mvar|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 {{mvar|S}} from the circle's center and is called the absolute value of the power of a point; more precisely, it can be stated that:

$$|AS|\backslash cdot|SC|\; =\; |BS|\backslash cdot|SD|\; =\; r^2-d^2,$$

where {{mvar|r}} is the radius of the circle, and {{mvar|d}} is the distance between the center of the circle and the intersection point {{mvar|S}}. This property follows directly from applying the chord theorem to a third chord (a diameter) going through {{mvar|S}} and the circle's center {{mvar|M}} (see drawing).

The theorem can be proven using similar triangles (via the inscribed-angle theorem). Consider the angles of the triangles {{math|△ASD}} and {{math|△BSC}}:

$$\backslash begin\{align\}$$

\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 {{math|△ASD}} and {{math|△BSC}} are similar and therefore

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

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 a point theorem.