# Intersecting Chord Theorem

## Contents

## Theorem

Let $AC$ and $BD$ both be chords of the same circle.

Let $AC$ and $BD$ intersect at $E$.

Then $AE \cdot EC = DE \cdot EB$.

In the words of Euclid:

*If in a circle two straight lines cut one another, the rectangle contained by the segments of the one is equal to the rectangle contained by the segments of the other.*

(*The Elements*: Book $\text{III}$: Proposition $35$)

## Proof 1

Let $AC$ and $BD$ be intersecting chords of circle $ABCD$.

Let the point of intersection be $E$.

If $E$ is the center of $ABCD$ the solution is trivial, as $AE = EC = BE = ED$ and so $AE \cdot EC = BE \cdot ED$.

Otherwise, let $F$ be the center of $ABCD$.

Let $FG$ be drawn perpendicular to $AC$, and $FH$ be drawn perpendicular to $BD$.

From Conditions for Diameter to be Perpendicular Bisector, $G$ bisects $AC$ and $H$ bisects $BD$.

So $AG = GC$ and $BH = HD$.

From Difference of Two Squares we have that $AE \cdot EC + EG^2 = GC^2$.

Let us add $GF^2$ to these.

So $AE \cdot EC + EG^2 + GF^2 = GC^2 + GF^2$.

But from Pythagoras's Theorem we have that:

- $GC^2 + GF^2 = CF^2$
- $EG^2 + GF^2 = EF^2$

So:

- $AE \cdot EC + EF^2 = CF^2$

Using the same construction, we have that:

- $DE \cdot EB + EF^2 = BF^2$

But $BF = CF$ as both are the radius of the circle $ABCD$.

That gives us:

- $AE \cdot EC + EF^2 = DE \cdot EB + EF^2$

It follows that $AE \cdot EC = DE \cdot EB$

$\blacksquare$

## Proof 2

Join $A$ with $B$ and $C$ with $D$, as shown in this diagram:

Then we have:

\(\displaystyle \angle AEB\) | \(\cong\) | \(\displaystyle \angle DEC\) | $\quad$ Two Straight Lines make Equal Opposite Angles | $\quad$ | |||||||||

\(\displaystyle \angle BAE\) | \(\cong\) | \(\displaystyle \angle CDE\) | $\quad$ Angles in Same Segment of Circle are Equal | $\quad$ |

By Triangles with Two Equal Angles are Similar we have $\triangle AEB \sim \triangle DEC$.

Thus:

\(\displaystyle \frac{AE}{EB}\) | \(=\) | \(\displaystyle \frac{DE}{EC}\) | $\quad$ | $\quad$ | |||||||||

\(\displaystyle \implies \ \ \) | \(\displaystyle AE \cdot EC\) | \(=\) | \(\displaystyle DE \cdot EB\) | $\quad$ | $\quad$ |

$\blacksquare$

## Also known as

This result is also known as the **chord theorem**.