Straight Line Commensurable with Apotome of Medial Straight Line

Proof

 * Euclid-X-103.png

Let $AB$ be an apotome of a medial straight line.

Let $CD$ be commensurable in length with $AB$.

It is to be demonstrated that:
 * $CD$ is an apotome of a medial straight line

and:
 * the order of $CD$ is the same as the order of $AB$.

Let $BE$ be the annex of $CD$.

Therefore by definition of apotome of a medial straight line:
 * $AE$ and $EB$ are medial straight lines which are commensurable in square only.

From, let it be contrived that:
 * $BE : DF = AB : CD$

From :
 * $AE : CF = AB : CD$

and so from :
 * $AE : CF = BE : DF$

But $AB$ is commensurable in length with $CD$.

Therefore from :
 * $AE$ is commensurable in length with $CF$

and:
 * $BE$ is commensurable in length with $DF$.

We have that $AE$ and $EB$ are medial straight lines which are commensurable in square only.

From :
 * $CF$ and $FD$ are medial straight lines.

Therefore from :
 * $CF$ and $FD$ are medial straight lines which are commensurable in square only.

Thus $CD$ is an apotome of a medial straight line.

It remains to be shown that the order of $CD$ is the same as the order of $AB$.

We have that:
 * $AE : CF = BE : DF$

So from :
 * $AE : EB = CF : FD$

Therefore:
 * $AE^2 : AE \cdot EB = CF^2 : CF \cdot FD$

But $AE^2$ is commensurable with $CF^2$.

Therefore from:

and:

we have that:
 * $AE \cdot EB$ is commensurable with $CF \cdot FD$.

Therefore by :
 * if $AE \cdot EB$ is rational, then $CF \cdot FD$ is rational.

From :
 * if $AE \cdot EB$ is medial, then $CF \cdot FD$ is medial.

Thuis $CD$ is an apotome of a medial straight line of the same as the order as $AB$.