Theorem. Two similar segments of circles which do not coincide cannot be constructed on the same chord (AB), and on the same side of that chord.
Dem.—If possible, let ACB, ADB, be two similar segments constructed on the same side of AB. Take any point D in the inner one. Join AD, and produce it to meet the outer one in C. Join BC, BD. Then since the segments are similar, the angle ADB is equal to ACB (Def. x.), which is impossible [I. xvi.]. Hence two similar segments not coinciding cannot be described on the same chord and on the same side of it.