Construction of Minor is Unique

Proof

 * Euclid-X-79.png

Let $AB$ be a minor.

Let $BC$ be added to $AB$ such that:
 * $AC$ and $CB$ are incommensurable in square
 * $AC^2 + CB^2$ is rational
 * $2 \cdot AC \cdot CB$ is a medial rectangle.

It is to be proved that no other straight line can be added to $AB$ which fulfils these conditions.

Suppose $BD$, different from $BC$, can be added to $AB$ such that:
 * $AD$ and $DB$ are incommensurable in square
 * $AD^2 + DB^2$ is rational
 * $2 \cdot AD \cdot DB$ is a medial rectangle.

From :
 * $AD^2 + DB^2 - 2 \cdot AD \cdot DB = AC^2 + CB^2 - 2 \cdot AC \cdot CB = AB^2$

Therefore:
 * $AD^2 + DB^2 - AC^2 + CB^2 = 2 \cdot AD \cdot DB - 2 \cdot AC \cdot CB$

But :$AD^2 + DB^2$ and $AC^2 + CB^2$ are both rational.

Therefore $AD^2 + DB^2 - AC^2 + CB^2$ is rational.

Therefore $2 \cdot AD \cdot DB - 2 \cdot AC \cdot CB$ is rational.

But by definition of medial area, this is impossible.

The result follows.