Proof: By Euclid
(related to Lemma: Lem. 12.04: Proportion of Sizes of Tetrahedra divided into Two Similar Tetrahedra and Two Equal Prisms)
- For, in the same figure, let perpendiculars have been conceived (drawn) from (points) $G$ and $H$ to the planes $ABC$ and $DEF$ (respectively).
- These clearly turn out to be equal, on account of the pyramids being assumed (to be) of equal height.
- And since two straight lines, $GC$ and the perpendicular from $G$, are cut by the parallel planes $ABC$ and $PMN$ they will be cut in the same ratios [Prop. 11.17].
- And $GC$ was cut in half by the plane $PMN$ at $N$.
- Thus, the perpendicular from $G$ to the plane $ABC$ will also be cut in half by the plane $PMN$.
- So, for the same (reasons), the perpendicular from $H$ to the plane $DEF$ will also be cut in half by the plane $STU$.
- And the perpendiculars from $G$ and $H$ to the planes $ABC$ and $DEF$ (respectively) are equal.
- Thus, the perpendiculars from the triangles $PMN$ and $STU$ to $ABC$ and $DEF$ (respectively, are) also equal.
- Thus, the prisms whose bases are triangles $LOC$ and $RVF$, and opposite (sides) $PMN$ and $STU$ (respectively), [are] of equal height.
- And, hence, the parallelepiped solids described on the aforementioned prisms [are] of equal height and (are) to one another as their bases [Prop. 11.32].
- Likewise, the halves (of the solids) [Prop. 11.28].
- Thus, as base $LOC$ is to base $RVF$, so the aforementioned prisms (are) to one another.
- (Which is) the very thing it was required to show.
Adapted from (subject to copyright, with kind permission)
- Fitzpatrick, Richard: Euclid's "Elements of Geometry"