Intersecting chords theorem

The intersecting chords theorem or just chord theorem is a statement in elementary geometry that describes a relation of the four line segments created by two intersecting chords in a circle. It states that the products of the lengths of the line segments on each chord are equal.

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, 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 theorem can be proven using similar triangles. 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 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.

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