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Proof: By Euclid

(related to Proposition: 2.11: Constructing the Golden Ratio of a Segment)

• For let the square $ABDC$ have been described on $AB$ [Prop. 1.46], and let $AC$ have been cut in half at point $E$ [Prop. 1.10], and let $BE$ have been joined.
• And let $CA$ have been drawn through to (point) $F$, and let $EF$ be made equal to $BE$ [Prop. 1.3].
• And let the square $FH$ have been described on $AF$ [Prop. 1.46], and let $GH$ have been drawn through to (point) $K$.
• I say that $AB$ has been cut at $H$ such as to make the rectangle contained by $AB$ and $BH$ equal to the square on $AH$.
• For since the straight line $AC$ has been cut in half at $E$, and $FA$ has been added to it, the rectangle contained by $CF$ and $FA$, plus the square on $AE$, is thus equal to the square on $EF$ [Prop. 2.6].
• And $EF$ (is) equal to $EB$.
• Thus, the (rectangle contained) by $CF$ and $FA$, plus the (square) on $AE$, is equal to the (square) on $EB$.
• But, the (sum of the squares) on $BA$ and $AE$ is equal to the (square) on $EB$.
• For the angle at $A$ (is) a right angle [Prop. 1.47].
• Thus, the (rectangle contained) by $CF$ and $FA$, plus the (square) on $AE$, is equal to the (sum of the squares) on $BA$ and $AE$.
• Let the square on $AE$ have been subtracted from both.
• Thus, the remaining rectangle contained by $CF$ and $FA$ is equal to the square on $AB$.
• And $FK$ is the (rectangle contained) by $CF$ and $FA$.
• For $AF$ (is) equal to $FG$.
• And $AD$ (is) the (square) on $AB$.
• Thus, the (rectangle) $FK$ is equal to the (square) $AD$.
• Let (rectangle) $AK$ have been subtracted from both.
• Thus, the remaining (square) $FH$ is equal to the (rectangle) $HD$.
• And $HD$ is the (rectangle contained) by $AB$ and $BH$.
• For $AB$ (is) equal to $BD$.
• And $FH$ (is) the (square) on $AH$.
• Thus, the rectangle contained by $AB$ and $BH$ is equal to the square on $HA$.
• Thus, the given straight line $AB$ has been cut at (point) $H$ such as to make the rectangle contained by $AB$ and $BH$ equal to the square on $HA$.
• (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"