Proof
Apply one segment onto the other via superposition (the device used
in I.4): the equal chords coincide, and the equal inscribed angles
(Definition III.11) force the arcs to coincide as well. By III.23,
two similar segments on the same chord on the same side cannot
differ; hence the segments are equal.
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Full neighborhood
Depends on (3)
- III.23Proposition III.23On the same straight line there cannot be constructed two similar and unequal segments of circles on the same side.
- I.4Proposition I.4If two triangles have two sides equal to two sides respectively, and have the angles contained by the equal straight…
- III.11Definition III.11Similar segments of circles are those which admit equal angles, or in which the angles are equal to one another.
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