In equal circles equal angles stand on equal circumferences whether they stand at the centers or at the circumferences.

Let *ABC* and *DEF* be equal circles, and in them let there be equal angles, namely at the centers the angles *BGC* and *EHF,* and at the circumferences the angles *BAC* and *EDF.*

I say that the circumference *BKC* equals the circumference *ELF.*

Join *BC* and *EF.*

Now, since the circles *ABC* and *DEF* are equal, the radii are equal.

Thus the two straight lines *BG* and *GC* equal the two straight lines *EH* and *HF,* and the angle at *G* equals the angle at *H,* therefore the base *BC* equals the base *EF.*

And, since the angle at *A* equals the angle at *D,* the segment *BAC* is similar to the segment *EDF,* and they are upon equal straight lines.

But similar segments of circles on equal straight lines equal one another, therefore the segment *BAC* equals *EDF.* But the whole circle *ABC* also equals the whole circle *DEF,* therefore the remaining circumference *BKC* equals the circumference *ELF.*

Therefore *in equal circles equal angles stand on equal circumferences whether they stand at the centers or at the circumferences.*

Q.E.D.