Intersecting Chord Theorem

When two chords intersect each other inside a circle, the products of their segments are equal.

A.B = C.D

It is a little easier to see this in the diagram on the right. Each chord is cut into two segments at the point of where they intersect. One chord is cut into two line segments A and B. The other into the segments C and D.

This theorem states that A×B is always equal to C×D no matter where the chords are.

In the figure below, drag the orange dots around to reposition the chords. As long as they intersect inside the circle, you can see from the calculations that the theorem is always true. The two products are always the same.

(Note: Because the lengths are rounded off for clarity, the calculations will be slightly off if you enter the displayed values into your calculator).

A Practical use

When making doors or windows with curved tops we need to find the radius of the arch so we can lay them out with compasses. See Radius of an Arc for a way to do this using the Intersecting Chords Theorem.

Other circle topics


Equations of a circle

Angles in a circle