On the same straight line there cannot be constructed two similar and unequal segments of circles on the same side.

For, if possible, on the same straight line *AB* let two similar and unequal segments of circles *ACB* and *ADB* be constructed on the same side. Draw *ACD* through, and join *CB* and *DB.*

Then, since the segment *ACB* is similar to the segment *ADB,* and similar segments of circles are those which admit equal angles, the angle *ACB* equals the angle *ADB,* the exterior to the interior, which is impossible.

Therefore *on the same straight line there cannot be constructed two similar and unequal segments of circles on the same side.*

Q.E.D.