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Commentary on Proposition XII.7¶

If we denote by C-ABD a pyramid with vertex C and base ABD, Euclid’s argument is easily followed thus.

The $\square$ ABED being bisected by BD, (pyramid C-ABD) $=$ (pyramid C-DEB) [XII. 5] $\equiv$ (pyramid D-EBC).

And, the $\square$ EBCF being bisected by EC, (pyramid D-EBC) $=$ (pyramid D-ECF).

Thus (pyramid C-ABD) $=$ (pyramid D-EBC) $=$ (pyramid D-ECF), and these three pyramids make up the whole prism, so that each is one-third of the prism.

And, since (pyramid C-ABD) $\equiv$ (pyramid D-ABC), (pyramid D-ABC) $= \frac{1}{3}$ (prism ABC, DEF).