Geometry theorem relating the line segments created by intersecting chords in a circle
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:
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, D lie on a common circle. Or in other words, if the diagonals of a quadrilateralABCD 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 pointS 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).
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