◀ ▲ ▶Branches / Geometry / Elements-euclid / Book--6-similar-figures / Proof: By Euclid

Proof: By Euclid

(related to Proposition: 6.29: Construction of Parallelogram Equal to Given Figure Exceeding a Parallelogram)

• Let $AB$ have been cut in half at (point) $E$ [Prop. 1.10], and let the parallelogram $BF$, (which is) similar, and similarly laid out, to $D$, have been described on $EB$ [Prop. 6.18].
• And let (parallelogram) $GH$ have been constructed (so as to be) both similar, and similarly laid out, to $D$, and equal to the sum of $BF$ and $C$ [Prop. 6.25].
• And let $KH$ correspond to $FL$, and $KG$ to $FE$.
• And since (parallelogram) $GH$ is greater than (parallelogram) $FB$, $KH$ is thus also greater than $FL$, and $KG$ than $FE$.
• Let $FL$ and $FE$ have been produced, and let $FLM$ be (made) equal to $KH$, and $FEN$ to $KG$ [Prop. 1.3].
• And let (parallelogram) $MN$ have been completed.
• Thus, $MN$ is equal and similar to $GH$.
• But, $GH$ is similar to $EL$.
• Thus, $MN$ is also similar to $EL$ [Prop. 6.21].
• $EL$ is thus about the same diagonal as $MN$ [Prop. 6.26].
• Let their (common) diagonal $FO$ have been drawn, and let the (remainder of the) figure have been described.
• And since (parallelogram) $GH$ is equal to (parallelogram) $EL$ and (figure) $C$, but $GH$ is equal to (parallelogram) $MN$, $MN$ is thus also equal to $EL$ and $C$.
• Let $EL$ have been subtracted from both.
• Thus, the remaining gnomon $XWV$ is equal to (figure) $C$.
• And since $AE$ is equal to $EB$, (parallelogram) $AN$ is also equal to (parallelogram) $NB$ [Prop. 6.1], that is to say, (parallelogram) $LP$ [Prop. 1.43].
• Let (parallelogram) $EO$ have been added to both.
• Thus, the whole (parallelogram) $AO$ is equal to the gnomon $VWX$.
• But, the gnomon $VWX$ is equal to (figure) $C$.
• Thus, (parallelogram) $AO$ is also equal to (figure) $C$.
• Thus, the parallelogram $AO$, equal to the given rectilinear figure $C$, has been applied to the given straight line $AB$, overshooting by the parallelogrammic figure $QP$ which is similar to $D$, since $PQ$ is also similar to $EL$ [Prop. 6.24].
• (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"