Construction of First Apotome of Medial is Unique

Proof

 * Euclid-X-79.png

Let $AB$ be the first apotome of a medial straight line.

Let $BC$ be added to $AB$ such that:
 * $AC$ and $CB$ are medial straight lines
 * $AC$ and $CB$ are commensurable in square only
 * $AC \cdot CB$ is a rational rectangle.

It is to be proved that no other medial straight line can be added to $AB$ which is commensurable in square only with the whole and which contains with the whole a rational rectangle.

Suppose $BD$ can be added to $AB$ so as to fulfil the conditions stated.

Then by definition of the first apotome of a medial straight line, $AD$ and $DB$ are such that:
 * $AD$ and $DB$ are medial straight lines
 * $AD$ and $DB$ are commensurable in square only
 * $AD \cdot DB$ is a rational 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 $AD \cdot DB$ and $AC \cdot CB$ are both rational.

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

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

But from:

and:

both $AD^2 + DB^2$ and $AC^2 + CB^2$ is medial.

By this cannot happen.

Therefore only one medial straight line can be added to $AB$ which is commensurable in square only with the whole and which contains with the whole a rational rectangle.