Equal Chords in Circle
Theorem
In a circle, equal chords are equally distant from the center, and chords that are equally distant from the center are equal in length.
In the words of Euclid:
- In a circle, equal straight lines are equally distant from the centre, and those which are equally distant from the centre are equal in length.
(The Elements: Book $\text{III}$: Proposition $14$)
Proof
Let $ABDC$ be a circle, and let $AB$ and $CD$ be equal chords on it.
From Finding Center of Circle, let $E$ be the center of $ABDC$.
Construct $EF$, $EG$ perpendicular to $AB$ and $CD$, and join $AE$ and $EC$.
From Conditions for Diameter to be Perpendicular Bisector, $EF$ bisects $AD$.
So $AF = FB$ and so $AB = 2 AF$.
Similarly $CD = 2CG$.
As $AB = CD$ it follows that $AE = CG$.
Since $AE = EC$ the square on $AE$ equals the square on $EC$.
We have that $\angle AFE$ is a right angle.
So from Pythagoras's Theorem the square on $AE$ equals the sum of the squares on $AF$ and $FE$.
Similarly, the square on $EC$ equals the sum of the squares on $EG$ and $GC$.
Thus the square on $EF$ equals the square on $EG$.
So $EF = EG$.
So from Definition III: 4, chords are at equal distance from the center when the perpendiculars drawn to them from the center are equal.
So $AB$ and $CD$ are at equal distance from the center.
$\Box$
Now suppose that $AB$ and $CD$ are at equal distance from the center.
That is, let $EF = EG$.
Using the same construction as above, it is proved similarly that $AB = 2AF$ and $CD = 2CG$.
Since $AE = CE$, the square on $AE$ equals the square on $CE$.
From Pythagoras's Theorem:
- the square on $AE$ equals the sum of the squares on $AF$ and $FE$
- the square on $EC$ equals the sum of the squares on $EG$ and $GC$.
So the sum of the squares on $AF$ and $FE$ equals the sum of the squares on $EG$ and $GC$.
But the square on $FE$ equals the square on $EG$.
So the square on $AF$ equals the square on $CG$, and so $AF = CG$.
As $AB = 2AF$ and $CD = 2CG$ it follows that $AB = CD$.
Hence the result.
$\blacksquare$
Historical Note
This proof is Proposition $14$ of Book $\text{III}$ of Euclid's The Elements.
Sources
- 1926: Sir Thomas L. Heath: Euclid: The Thirteen Books of The Elements: Volume 2 (2nd ed.) ... (previous) ... (next): Book $\text{III}$. Propositions