On the same straight line there cannot be constructed two similar and unequal segments of circles on the same side.
Proof
Suppose two similar but unequal segments are constructed on the
same chord AB on the same side. Pick a point C on the smaller
segment's arc. The inscribed angle ∠ACB in the smaller
segment equals (by Definition III.11) the inscribed angle in the
larger segment, since the segments are similar. But the larger
segment's arc lies entirely outside the smaller's arc (different
sizes, same chord, same side), so an inscribed angle at a point
C on the smaller arc as viewed from a point on the larger arc
would have to differ from the corresponding inscribed angle in the
larger segment (by III.21 they all agree within each segment) —
the configurations are incompatible. The two segments must
coincide.