# Segment on Given Base Unique

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## Theorem

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 $23$)

## Proof

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 Greater than Internal Opposite is impossible.

$\blacksquare$

## Historical Note

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

## Sources

- 1926: Sir Thomas L. Heath:
*Euclid: The Thirteen Books of The Elements: Volume 2*(2nd ed.) ... (previous) ... (next): Book $\text{III}$. Propositions