Proof: By Euclid
(related to Proposition: 2.07: Sum of Squares)
- For let the square $ADEB$ have been described on $AB$ [Prop. 1.46], and let the (rest of) the figure have been drawn.
- Therefore, since (rectangle) $AG$ is equal to (rectangle) $GE$ [Prop. 1.43], let the (square) $CF$ have been added to both.
- Thus, the whole (rectangle) $AF$ is equal to the whole (rectangle) $CE$.
- Thus, (rectangle) $AF$ plus (rectangle) $CE$ is double (rectangle) $AF$.
- But, (rectangle) $AF$ plus (rectangle) $CE$ is the gnomon $KLM$, and the square $CF$.
- Thus, the gnomon $KLM$, and the square $CF$, is double the (rectangle) $AF$.
- But double the (rectangle) $AF$ is also twice the (rectangle contained) by $AB$ and $BC$.
- For $BF$ (is) equal to $BC$.
- Thus, the gnomon $KLM$, and the square $CF$, are equal to twice the (rectangle contained) by $AB$ and $BC$.
- Let $DG$, which is the square on $AC$, have been added to both.
- Thus, the gnomon $KLM$, and the squares $BG$ and $GD$, are equal to twice the rectangle contained by $AB$ and $BC$, and the square on $AC$.
- But, the gnomon $KLM$ and the squares $BG$ and $GD$ is (equivalent to) the whole of $ADEB$ and $CF$, which are the squares on $AB$ and $BC$ (respectively).
- Thus, the (sum of the) squares on $AB$ and $BC$ is equal to twice the rectangle contained by $AB$ and $BC$, and the square on $AC$.
- Thus, if a straight line is cut at random then the sum of the squares on the whole (straight line), and one of the pieces (of the straight line), is equal to twice the rectangle contained by the whole, and the said piece, and the square on the remaining piece.
- (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"
