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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.
Circle, two chords AB and CD that intersect at E.

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


This theorem is also known as the Intersecting Chords Theorem.

Proof

Consider the auxiliary segments and

Circle, two chords AB and CD that intersect at E. Chords AC and BD drawn in blue and dashed.

Because and are inscribed angles that intercept the same arc they are congruent angles. Similarly, and are inscribed angles that intercept the same arc Therefore, they are also congruent angles.

Circle, two chords AB and CD that intersect at E. Both, intercepted angles and arcs labeled. Angle CAB is congruent to angle CDB. Angle ACD is congruent to angle ABD.
Consequently, by the Angle-Angle Similarity Theorem, and are similar triangles. This relationship means that the following proportion can be set.
Finally, by cross multiplying, the desired result is obtained.


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