Commentary on Proposition XII.3¶
Back to Proposition XII.3.
We will denote a pyramid with vertex D and base ABC by D(ABC) or D-ABC and the triangular prism with triangles GCF, HLK for bases by (GCF, HLK).
The following are the steps of the proof.
To prove pyramid H(AEG) equal and similar to pyramid D(HKL).
Since sides of $\triangle$ DAB are bisected at H, E, K, HE $\parallel$ DB, and HK $\parallel$ AB. Hence HK $=$ EB $=$ EA, HE $=$ KB $=$ DK. Therefore (1) $\triangle\text{s}$ HAE, DHK are equal and similar. Similarly (2) $\triangle\text{s}$ HAG, DHL are equal and similar. Again, LH, HK are respectively $\parallel$ to GA, AE in a different plane; therefore $\angle$ GAE $= \angle$ LHK.
And LH, HK are respectively equal to GA, AE. Therefore (3) $\triangle\text{s}$ GAE, LHK are equal and similar. Similarly (4) $\triangle\text{s}$ HGE, DLK are equal and similar. Therefore [XI. Def. 10] the pyramids H(AEG) and D(HKL) are equal and similar.
II. To prove the pyramid D(HKL) similar to the pyramid D(ABC). (1) The $\triangle\text{s}$ DHK, DAB are equiangular and therefore similar.
Similarly (2) $\triangle\text{s}$ DLH, DCA are similar, as also (3) the $\triangle\text{s}$ DLK, DCB. Again, BA, AC are respectively parallel to KH, HL in a different plane; therefore $\angle$ BAC $= \angle$ KHL. And BA $:$ AC $=$ KH $:$ HL. Therefore (4) $\triangle\text{s}$ BAC, KHL are similar. Consequently the pyramid D(ABC) is similar to the pyramid D(HKL), and therefore also to the pyramid H(AEG).
To prove prism (GCF, HLK) equal to prism (HGE, KFB).
The prisms may be regarded as having the same _height_ (the distance between the planes HKL, ABC) and having for bases (1) the $\triangle$ CGF and (2) the $\square$ EBFG, which is the double of the $\triangle$ CGF. Therefore, by XI. 39, the prisms are equal.
To prove the prisms greater than the small pyramids.
Prism (HGE, KFB) is clearly greater than pyramid K(EFB) and therefore greater than pyramid H(AEG). Therefore each of the prisms is greater than each of the small pyramids; and the sum of the two prisms is greater than the sum of the two small pyramids, which, with the two prisms, make up the whole pyramid.