Segment on Given Base Unique

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In the words of Euclid:

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

(The Elements: Book $\text{III}$: Proposition $23$)



Suppose it were possible to construct two similar and unequal segments $ACB$ and $ADB$ on the same base $AB$.

Let $ACD$ be drawn through, and join $CB$ and $CB$.

We have by hypothesis that segment $ACD$ is similar to $ADB$.

We also have by Book $\text{III}$ Definition $11$: Similar Segments that similar segments admit equal angles.

So $\angle ACB = \angle ADB$, which from External Angle of Triangle is Greater than Internal Opposite is impossible.


Historical Note

This proof is Proposition $23$ of Book $\text{III}$ of Euclid's The Elements.