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Proof: By Euclid

(related to Proposition: 6.17: Rectangles Contained by Three Proportional Straight Lines)

• Let $A$, $B$ and $C$ be three proportional straight lines, (such that) as $A$ (is) to $B$, so $B$ (is) to $C$.
• I say that the rectangle contained by $A$ and $C$ is equal to the square on $B$.
  • Let $D$ be made equal to $B$ [Prop. 1.3].
  • And since as $A$ is to $B$, so $B$ (is) to $C$, and $B$ (is) equal to $D$, thus as $A$ is to $B$, (so) $D$ (is) to $C$.
  • And if four straight lines are proportional then the [rectangle] contained by the (two) outermost is equal to the rectangle contained by the middle (two) [Prop. 6.16].
  • Thus, the (rectangle contained) by $A$ and $C$ is equal to the (rectangle contained) by $B$ and $D$.
  • But, the (rectangle contained) by $B$ and $D$ is the (square) on $B$.
  • For $B$ (is) equal to $D$.
  • Thus, the rectangle contained by $A$ and $C$ is equal to the square on $B$.
• And so, let the (rectangle contained) by $A$ and $C$ be equal to the (square) on $B$.
• I say that as $A$ is to $B$, so $B$ (is) to $C$.
  • For, with the same construction, since the (rectangle contained) by $A$ and $C$ is equal to the (square) on $B$.
  • But, the (square) on $B$ is the (rectangle contained) by $B$ and $D$.
  • For $B$ (is) equal to $D$.
  • The (rectangle contained) by $A$ and $C$ is thus equal to the (rectangle contained) by $B$ and $D$.
  • And if the (rectangle contained) by the (two) outermost is equal to the (rectangle contained) by the middle (two) then the four straight lines are proportional [Prop. 6.16].
  • Thus, as $A$ is to $B$, so $D$ (is) to $C$.
  • And $B$ (is) equal to $D$.
  • Thus, as $A$ (is) to $B$, so $B$ (is) to $C$.
  • Thus, if three straight lines are proportional then the rectangle contained by the (two) outermost is equal to the square on the middle (one).
  • And if the rectangle contained by the (two) outermost is equal to the square on the middle (one) then the three straight lines will be proportional.
• (Which is) the very thing it was required to show.
  ∎

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References

Adapted from (subject to copyright, with kind permission)

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