Intersecting_chords_theorem
Intersecting chords theorem
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 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 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
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.