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Proof: By Euclid

(related to Proposition: 2.14: Constructing a Square from a Rectilinear Figure)

• For let the right-angled parallelogram $BD$, equal to the rectilinear figure $A$, have been constructed [Prop. 1.45].
• Therefore, if $BE$ is equal to $ED$ then that (which) was prescribed has taken place.
• For the square $BD$, equal to the rectilinear figure $A$, has been constructed.
• And if not, then one of the (straight lines) $BE$ or $ED$ is greater (than the other).
• Let $BE$ be greater, and let it have been produced to $F$, and let $EF$ be made equal to $ED$ [Prop. 1.3].
• And let $BF$ have been cut in half at (point) $G$ [Prop. 1.10].
• And, with center $G$, and radius one of the (straight lines) $GB$ or $GF$, let the semicircle $BHF$ have been drawn.
• And let $DE$ have been produced to $H$, and let $GH$ have been joined.
• Therefore, since the straight line $BF$ has been cut - equally at $G$, and unequally at $E$ - the rectangle contained by $BE$ and $EF$, plus the square on $EG$, is thus equal to the square on $GF$ [Prop. 2.5].
• And $GF$ (is) equal to $GH$.
• Thus, the (rectangle contained) by $BE$ and $EF$, plus the (square) on $GE$, is equal to the (square) on $GH$.
• And the (sum of the) squares on $HE$ and $EG$ is equal to the (square) on $GH$ [Prop. 1.47].
• Thus, the (rectangle contained) by $BE$ and $EF$, plus the (square) on $GE$, is equal to the (sum of the squares) on $HE$ and $EG$.
• Let the square on $GE$ have been taken from both.
• Thus, the remaining rectangle contained by $BE$ and $EF$ is equal to the square on $EH$.
• But, $BD$ is the (rectangle contained) by $BE$ and $EF$.
• For $EF$ (is) equal to $ED$.
• Thus, the parallelogram $BD$ is equal to the square on $HE$.
• And $BD$ (is) equal to the rectilinear figure $A$.
• Thus, the rectilinear figure $A$ is also equal to the square (which) can be described on $EH$.
• Thus, a square - (namely), that (which) can be described on $EH$ - has been constructed, equal to the given rectilinear figure $A$.
• (Which is) the very thing it was required to do.
  ∎

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References

Adapted from (subject to copyright, with kind permission)

1. Fitzpatrick, Richard: Euclid's "Elements of Geometry"