◀ ▲ ▶Branches / Geometry / Elements-euclid / Book-11-elementary-stereometry / Proof: By Euclid

Proof: By Euclid

(related to Proposition: Prop. 11.24: Opposite Planes of Solid contained by Parallel Planes are Equal Parallelograms)

• For let the solid (figure) $CDHG$ have been contained by the parallel planes $AC$, $GF$, and $AH$, $DF$, and $BF$, $AE$.

• I say that its opposite planes are both equal and parallelogrammic.

• For since the two parallel planes $BG$ and $CE$ are cut by the plane $AC$, their common sections are parallel [Prop. 11.16].

• Thus, $AB$ is parallel to $DC$.
• Again, since the two parallel planes $BF$ and $AE$ are cut by the plane $AC$, their common sections are parallel [Prop. 11.16].
• Thus, $BC$ is parallel to $AD$.
• And $AB$ was also shown (to be) parallel to $DC$.
• Thus, $AC$ is a parallelogram.
• So, similarly, we can also show that $DF$, $FG$, $GB$, $BF$, and $AE$ are each parallelograms.
• Let $AH$ and $DF$ have been joined.
• And since $AB$ is parallel to $DC$, and $BH$ to $CF$, so the two (straight lines) joining one another, $AB$ and $BH$, are parallel to the two straight lines joining one another, $DC$ and $CF$ (respectively), not (being) in the same plane.
• Thus, they will contain equal angles [Prop. 11.10].
• Thus, angle $ABH$ (is) equal to (angle) $DCF$.
• And since the two (straight lines) $AB$ and $BH$ are equal to the two (straight lines) $DC$ and $CF$ (respectively) [Prop. 1.34], and angle $ABH$ is equal to angle $DCF$, the base $AH$ is thus equal to the base $DF$, and triangle $ABH$ is equal to triangle $DCF$ [Prop. 1.4].
• And parallelogram $BG$ is double (triangle) $ABH$, and parallelogram $CE$ double (triangle) $DCF$ [Prop. 1.34].
• Thus, parallelogram $BG$ (is) equal to parallelogram $CE$.
• So, similarly, we can show that $AC$ is also equal to $GF$, and $AE$ to $BF$.
• Thus, if a solid (figure) is contained by (six) parallel planes then its opposite planes are both equal and parallelogrammic.
• (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"