Theorem.
In equal circles (ACB, DFE), equal chords (AB, DE) divide the circumferences into arcs, which are equal each to each—that is, the lesser to the lesser, and the greater to the greater.
Dem.—If the equal chords be diameters, the Proposition is evident. If not, let O, H be the centres. Join AO, OB, DH, HE; then because the circles are equal their radii are equal (Def. i.). Hence the two triangles AOB, DHE have the sides AO, OB in one respectively equal to the sides DH, HE in the other, and the base AB is equal to DE (hyp.). Therefore [I. viii.] the angle AOB is equal to DHE. Hence the arc AGB is equal to DKE [xxvi.]; and since the whole circumference AGBC is equal to the whole circumference DKEF, the remaining arc ACB is equal to the remaining arc DFE.