(related to Proposition: 3.23: Segment on Given Base Unique)

- For, if possible, let the two similar and unequal segments of circles, $ACB$ and $ADB$, have been constructed on the same side of the same straight line $AB$.
- And let $ACD$ have been drawn through (the segments), and let $CB$ and $DB$ have been joined.

- Therefore, since segment $ACB$ is similar to segment $ADB$, and similar segments of circles are those accepting equal angles [Def. 3.11] , angle $ACB$ is thus equal to $ADB$, the external to the internal.
- The very thing is impossible [Prop. 1.16].
- Thus, two similar and unequal segments of circles cannot be constructed on the same side of the same straight line.∎

**Fitzpatrick, Richard**: Euclid's "Elements of Geometry"