Segments of Secants Theorem

Consider the auxiliary segments $\overline{AD}$ and $\overline{BC}.$

Notice that $\angle EBC$ and $\angle EDA$ 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 $\angle E\cong \angle E.$ 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. Finally, by cross multiplying, the desired result is obtained.