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Rule

Segments of Chords Theorem

If two chords intersect in the interior of a circle, then the product of the lengths of the segments of one chord is equal to the product of the lengths of the segments of the other chord.

Based on the diagram above, the relation below holds true.

$A E \cdot E B = C E \cdot E D$

This theorem is also known as the Intersecting Chords Theorem.

Proof

Consider the auxiliary segments $\overline{A C}$ and $\overline{B D} .$

Because $∠ C A B$ and $∠ C D B$ are inscribed angles that intercept the same arc $C B,$ they are congruent angles. Similarly, $∠ A C D$ and $∠ A B D$ are inscribed angles that intercept the same arc $A D .$ Therefore, they are also congruent angles.

Consequently, by the Angle-Angle Similarity Theorem,

△ A E C

and

△ D E B

are similar triangles. This relationship means that the following proportion can be set.

\frac{A E}{E D} = \frac{C E}{E B}

Finally, by cross multiplying, the desired result is obtained.

$A E \cdot E B = C E \cdot E D$

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