Proof: By Euclid
(related to Proposition: 6.30: Construction of the Inverse Golden Section)
- Let the square $BC$ have been described on $AB$ [Prop. 1.46], and let the parallelogram $CD$, equal to $BC$, have been applied to $AC$, overshooting by the figure $AD$ (which is) similar to $BC$ [Prop. 6.29].
- And $BC$ is a square.
- Thus, $AD$ is also a square.
- And since $BC$ is equal to $CD$, let (rectangle) $CE$ have been subtracted from both.
- Thus, the remaining (rectangle) $BF$ is equal to the remaining (square) $AD$.
- And it is also equiangular to it.
- Thus, the sides of $BF$ and $AD$ about the equal angles are reciprocally proportional [Prop. 6.14].
- Thus, as $FE$ is to $ED$, so $AE$ (is) to $EB$.
- And $FE$ (is) equal to $AB$, and $ED$ to $AE$.
- Thus, as $BA$ is to $AE$, so $AE$ (is) to $EB$.
- And $AB$ (is) greater than $AE$.
- Thus, $AE$ (is) also greater than $EB$ [Prop. 5.14].
- Thus, the straight line $AB$ has been cut in extreme and mean ratio at $E$, and $AE$ is its greater piece.
- (Which is) the very thing it was required to do.
Thank you to the contributors under CC BY-SA 4.0!
Adapted from (subject to copyright, with kind permission)
- Fitzpatrick, Richard: Euclid's "Elements of Geometry"