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Proof: By Euclid

(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.
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References

Adapted from (subject to copyright, with kind permission)

1. Fitzpatrick, Richard: Euclid's "Elements of Geometry"