Intersecting secants theorem

In Euclidean geometry, the intersecting secants theorem or just secant theorem describes the relation of line segments created by two intersecting secants and the associated circle.

${\displaystyle \triangle PAC\sim \triangle PBD}$

yields

${\displaystyle |PA|\cdot |PD|=|PB|\cdot |PC|}$

For two lines AD and BC that intersect each other at P and for which A, B, C, D all lie on the same circle, the following equation holds:

$${\displaystyle |PA|\cdot |PD|=|PB|\cdot |PC|}$$

The theorem follows directly from the fact that the triangles △PAC and △PBD are similar. They share ∠DPC and ∠ADB = ∠ACB as they are inscribed angles over AB. The similarity yields an equation for ratios which is equivalent to the equation of the theorem given above:

$${\displaystyle {\frac {PA}{PC}}={\frac {PB}{PD}}\Leftrightarrow |PA|\cdot |PD|=|PB|\cdot |PC|}$$

Next to the intersecting chords theorem and the tangent-secant theorem, the intersecting secants 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.