Proof: By Euclid
(related to Proposition: Prop. 12.04: Proportion of Sizes of Tetrahedra divided into Two Similar Tetrahedra and Two Equal Prisms)

- For since BO is equal to OC, and AL to LC, LO is thus parallel to AB, and triangle ABC similar to triangle LOC [Prop. 12.3].
- So, for the same (reasons), triangle DEF is also similar to triangle RVF.
- And since BC is double CO, and EF (double) FV, thus as BC (is) to CO, so EF (is) to FV.
- And the similar, and similarly laid out, rectilinear (figures) ABC and LOC have been described on BC and CO (respectively), and the similar, and similarly laid out, [rectilinear] (figures) DEF and RVF on EF and FV (respectively).
- Thus, as triangle ABC is to triangle LOC, so triangle DEF (is) to triangle RVF [Prop. 6.22].
- Thus, alternately, as triangle ABC is to [triangle] DEF, so [triangle] LOC (is) to triangle RVF [Prop. 5.16].
- But, as triangle LOC (is) to triangle RVF, so the prism whose base [is] [triangle]bookofproofs$6432 LOC, and opposite (plane) PMN, (is) to the prism whose base (is) triangle RVF, and opposite (plane) STU (see lemma).
- And, thus, as triangle ABC (is) to triangle DEF, so the prism whose base (is) triangle LOC, and opposite (plane) PMN, (is) to the prism whose base (is) triangle RVF, and opposite (plane) STU.
- And as the aforementioned prisms (are) to one another, so the prism whose base (is) parallelogram KBOL, and opposite (side) straight line PM, (is) to the prism whose base (is) parallelogram QEVR, and opposite (side) straight line ST [Prop. 11.39], [Prop. 12.3].
- Thus, also, (is) the (sum of the) two prisms - that whose base (is) parallelogram KBOL, and opposite (side) PM, and that whose base (is) LOC, and opposite (plane) PMN - to (the sum of) the (two) prisms - that whose base (is) QEVR, and opposite (side) straight line ST, and that whose base (is) triangle RVF, and opposite (plane) STU [Prop. 5.12].
- And, thus, as base ABC (is) to base DEF, so the (sum of the first) aforementioned two prisms (is) to the (sum of the second) aforementioned two prisms.
And, similarly, if pyramids PMNG and STUH are divided into two prisms, and two pyramids, as base PMN (is) to base STU, so (the sum of) the two prisms in pyramid PMNG will be to (the sum of) the two prisms in pyramid STUH.
- But, as base PMN (is) to base STU, so base ABC (is) to base DEF.
- For the triangles PMN and STU (are) equal to LOC and RVF, respectively.
- And, thus, as base ABC (is) to base DEF, so (the sum of) the four prisms (is) to (the sum of) the four prisms [Prop. 5.12].
- So, similarly, even if we divide the pyramids left behind into two pyramids and into two prisms, as base ABC (is) to base DEF, so (the sum of) all the prisms in pyramid ABCG will be to (the sum of) all the equal number of prisms in pyramid DEFH.
- (Which is) the very thing it was required to show.
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References
Adapted from (subject to copyright, with kind permission)
- Fitzpatrick, Richard: Euclid's "Elements of Geometry"