Proof: By Euclid
(related to Proposition: 3.29: Equal Arcs of Circles Subtended by Equal Straight Lines)
 For let the centers of the circles have been found [Prop. 3.1], and let them be (at) $K$ and $L$.
 And let $BK$, $KC$, $EL$, and $LF$ have been joined.
 And since the circumference $BGC$ is equal to the circumference $EHF$, the angle $BKC$ is also equal to (angle) $ELF$ [Prop. 3.27].
 And since the circles $ABC$ and $DEF$ are equal, their radii are also equal [Def. 3.1] .
 So the two (straight lines) $BK$, $KC$ are equal to the two (straight lines) $EL$, $LF$ (respectively).
 And they contain equal angles.
 Thus, the base $BC$ is equal to the base $EF$ [Prop. 1.4].
 Thus, in equal circles, equal straight lines subtend equal circumferences.
 (Which is) the very thing it was required to show.^{1}
∎
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References
Adapted from (subject to copyright, with kind permission)
 Fitzpatrick, Richard: Euclid's "Elements of Geometry"
Footnotes