Segments of Secants Theorem

Consider the auxiliary segments AD and BC.

Notice that ∠ EBC and ∠ EDA are inscribed angles that intercept the same arc. Therefore, by the Inscribed Angles Theorem, these two angles are congruent.

Additionally, by the Reflexive Property of Congruence, it can be stated that ∠ E ≅ ∠ E. This means that △ EDA and △ EBC have two pairs of corresponding congruent angles. Consequently, by the Angle-Angle Similarity Theorem, △ EDA and △ EBC are similar triangles. This allows to set the following proportion. EA/EC = ED/EB Finally, by cross multiplying, the desired result is obtained.