In equal circles equal straight lines cut off equal circumferences, the greater circumference equals the greater and the less equals the less.

Let *ABC* and *DEF* be equal circles, and in the circles let *AB* and *DE* be equal straight lines cutting off *ACB* and *DFE* as greater circumferences and *AGB* and *DHE* as lesser.

I say that the greater circumference *ACB* equals the greater circumference *DFE,* and the less circumference *AGB* equals *DHE.*

Take the centers *K* and *L* of the circles, and join *AK, KB, DL,* and *LE.*

Now, since the circles are equal, the radii are also equal, therefore the two sides *AK* and *KB* equal the two sides *DL* and *LE,* and the base *AB* equals the base *DE,* therefore the angle *AKB* equals the angle *DLE.*

But equal angles stand on equal circumferences when they are at the centers, therefore the circumference *AGB* equals *DHE.*

And the whole circle *ABC* also equals the whole circle *DEF,* therefore the remaining circumference *ACB* also equals the remaining circumference *DFE.*

Therefore *in equal circles equal straight lines cut off equal circumferences, the greater circumference equals the greater and the less equals the less.*

Q.E.D.