Chord Lies Inside its Circle

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Theorem

In the words of Euclid:

If on the circumference of a circle two points be taken at random, the straight line joining the points will fall within the circle.

(The Elements: Book $\text{III}$: Proposition $2$)


Proof

Euclid-III-2.png

Let $ABC$ be a circle where $A, B$ are two points taken at random on its circumference.


Suppose the straight line $AB$ does not lie entirely within the circle $ABC$.

Then it will lie wholly outside (otherwise it will cross the circumference somewhere, and so we can take the part of $AB$ which does lie outside).

Find the center $D$ of circle $ABC$.

Draw $DA$, $DB$ and $DE$, and let $F$ be the point where $DE$ crosses the circumference.

As $DA = DB$ then from Isosceles Triangle has Two Equal Angles it follows that $\angle DAE = \angle DBE$.

Since one side of $AEB$ of $\triangle DAE$ is produced, it follows from External Angle of Triangle Greater than Internal Opposite that $\angle DEB$ is greater than $\angle DAE$.

But $\angle DAE = \angle DBE$, so $\angle DEB$ is greater than $\angle DBE$.

But from Greater Angle of Triangle Subtended by Greater Side, $DB$ is greater than $DE$.

But $DB = DF$, so $DF$ is greater than $DE$.

But $DF$ is less than $DE$.

Thus we have a contradiction.


So the straight line $AB$ does not fall outside the circle.


Similarly, we can show that $AB$ does not lie on the circumference itself.


Hence the result.

$\blacksquare$


Historical Note

This theorem is Proposition $2$ of Book $\text{III}$ of Euclid's The Elements.


Sources