Geometry theorem relating line segments created by intersecting secants of a circle
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.
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:
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: