Segment on Given Base Unique

Theorem

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

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 definition 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 is Proposition 23 of Book III of Euclid's The Elements.