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

 * Euclid-III-23.png

Suppose it were possible to construct two 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.