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Segments of Secants Theorem

Rule

Segments of Secants Theorem

If two secants intersect in the exterior of a circle, then the product of the measures of one secant segment and its external secant segment is equal to the product of the measures of the other secant segment and its external secant segment.

Base on the diagram above, the following relation holds true.

This theorem is also known as the Secant Segments Theorem or the Intersecting Secants Theorem.

Proof

Consider the auxiliary segments and

Notice that because these two inscribed angles intercept the same arc,

The Reflexive Property of Congruence gives Then, and have two pairs of corresponding congruent angles.
Separate Triangles

Reset

In consequence, thanks to the Angle-Angle Similarity Theorem. This allows to set the following proportion. Finally, by cross multiplying, the desired result is obtained.