Theorem. Similar segments of circles (AEB, CFD) on equal chords (AB, CD) are equal to one another.
Dem.—Since the lines are equal, if AB be applied to CD, so that the point A will coincide with C, and the line AB with CD, the point B shall coincide with D; and because the segments are similar, they must coincide [xxiii.]. Hence they are equal.
This demonstration may be stated as follows:—Since the chords are equal, they are congruent; and therefore the segments, being similar, must be congruent.