Straight Lines Cut Off Equal Arcs in Equal Circles

Proof
Let $ABC$ and $DEF$ be equal circles.

Let $AB, DE$ be equal straight lines cutting off arcs $ACB$ and $DFE$ as the greater, and $AGB, DHE$ as lesser.


 * Euclid-III-28.png

Let $K$ and $L$ be the centers of the circles $ABC$ and $DEF$ respectively.

Let $AK, KB, DL, LE$ be joined.

Since the circles are equal, so are their radii.

So $AK = DL, KB = LE$.

As $AB = DE$ by hypothesis, from Triangle Side-Side-Side Equality it follows that $\angle AKB = \angle DLE$.

But from Angles on Equal Arcs are Equal the arc $AGB$ equals the arc $DHE$.

As the whole circles $ABC$ and $DEF$ are equal, the arc $ACB$ which remains also equals arc $DFE$.