Angles in equal circles have the same ratio as the circumferences on which they stand whether they stand at the centers or at the circumferences.

Let *ABC* and *DEF* be equal circles, and let the angles *BGC* and *EHF* be angles at their centers *G* and *H,* and the angles *BAC* and *EDF* angles at the circumferences.

I say that the circumference *BC* is to the circumference *EF* as the angle *BGC* is to the angle *EHF,* and as the angle *BAC* is to the angle *EDF.*

Make any number of consecutive circumferences *CK* and *KL* equal to the circumference *BC,* and any number of consecutive circumferences *FM, MN* equal to the circumference *EF,* and join *GK* and *GL* and *HM* and *HN.*

Then, since the circumferences *BC, CK,* and *KL* equal one another, the angles *BGC, CGK,* and *KGL* also equal one another. Therefore, whatever multiple the circumference *BL* is of *BC,* the angle *BGL* is also that multiple of the angle *BGC.*

For the same reason, whatever multiple the circumference *NE* is of *EF,* the angle *NHE* is also that multiple of the angle *EHF.*

If the circumference *BL* equals the circumference *EN,* then the angle *BGL* also equals the angle *EHN*; if the circumference *BL* is greater than the circumference *EN,* then the angle *BGL* is also greater than the angle *EHN*; and, if less, less.

There being then four magnitudes, two circumferences *BC* and *EF,* and two angles *BGC* and *EHF,* there have been taken, of the circumference *BC* and the angle *BGC* equimultiples, namely the circumference *BL* and the angle *BGL,* and of the circumference *EF* and the angle *EHF* equimultiples, namely the circumference *EN* and the angle *EHN.*

And it has been proved that, if the circumference *BL* is in excess of the circumference *EN,* the angle *BGL* is also in excess of the angle *EHN*; if equal, equal; and if less, less.

Therefore the circumference *BC* is to *EF* as the angle *BGC* is to the angle *EHF.*

But the angle *BGC* is to the angle *EHF* as the angle *BAC* is to the angle *EDF,* for they are doubles respectively.

Therefore also the circumference *BC* is to the circumference *EF* as the angle *BGC* is to the angle *EHF,* and the angle *BAC* to the angle *EDF.*

Therefore, *angles in equal circles have the same ratio as the circumferences on which they stand whether they stand at the centers or at the circumferences.*

Q.E.D.

In the *Elements* Euclid restricted his study of lengths of arcs to circles of the same radius. He did not compare arcs of different sized circles. Later, however, Archimedes did just that in his *Measurement of a Circle*.

This proposition is used in three consecutive propositions in Book XIII starting with XIII.8 to convert statements about arcs to statements about angles. Incidentally, all three have to do with regular pentagons inscribed in circles.