Proof: By Euclid
(related to Proposition: Prop. 10.105: Straight Line Commensurable with Minor Straight Line)
- For let the same things have been contrived (as in the former proposition).
- And since $AE$ and $EB$ are (straight lines which are) incommensurable in square [Prop. 10.76], $CF$ and $FD$ are thus also (straight lines which are) incommensurable in square [Prop. 10.13].
- Therefore, since as $AE$ is to $EB$, so $CF$ (is) to $FD$ [Prop. 5.12], [Prop. 5.16], thus also as the (square) on $AE$ is to the (square) on $EB$, so the (square) on $CF$ (is) to the (square) on $FD$ [Prop. 6.22].
- Thus, via composition, as the (sum of the squares) on $AE$ and $EB$ is to the (square) on $EB$, so the (sum of the squares) on $CF$ and $FD$ (is) to the (square) on $FD$ [Prop. 5.18], [also alternately].
- And the (square) on $BE$ is commensurable with the (square) on on $DF$ [Prop. 10.104].
- The sum of the squares on $AE$ and $EB$ (is) thus also commensurable with the sum of the squares on $CF$ and $FD$ [Prop. 5.16], [Prop. 10.11].
- And the sum of the (squares) on $AE$ and $EB$ is rational [Prop. 10.76].
- Thus, the sum of the (squares) on $CF$ and $FD$ is also [Def. 10.4] .
- Again, since as the (square) on $AE$ is to the (rectangle contained) by $AE$ and $EB$, so the (square) on $CF$ (is) to the (rectangle contained) by $CF$ and $FD$ [Prop. 10.21 lem.] , and the square on $AE$ (is) commensurable with the square on $CF$, the (rectangle contained) by $AE$ and $EB$ is thus also commensurable with the (rectangle contained) by $CF$ and $FD$.
- And the (rectangle contained) by $AE$ and $EB$ (is) medial [Prop. 10.76].
- Thus, the (rectangle contained) by $CF$ and $FD$ (is) also medial [Prop. 10.23 corr.] .
- $CF$ and $FD$ are thus (straight lines which are) incommensurable in square, making the sum of the squares on them rational, and the (rectangle contained) by them medial.
- Thus, $CD$ is a minor (straight line) [Prop. 10.76].
- (Which is) the very thing it was required to show.
∎
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References
Adapted from (subject to copyright, with kind permission)
- Fitzpatrick, Richard: Euclid's "Elements of Geometry"
