Equal Arcs of Circles Subtended by Equal Straight Lines

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

Let equal arcs $BGC$ and $EHF$ be cut off by the straight lines $BC$ and $EF$.


 * Euclid-III-29.png

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

Let $BK, KC, EL, LF$ be joined.

We have that the arcs $BGC$ and $EHF$ are equal.

So from Angles on Equal Arcs are Equal $\angle BKC = \angle ELF$.

Since circles $ABC$ and $DEF$ are equal, so are their radii.

So $BK = EL$ and $KC = LF$, and they contain equal angles.

So from Triangle Side-Angle-Side Equality base $BC$ is equal to base $EF$.