Kitap 12
6
And again componendo, as the base ABCDE is to the base ADE, so is the pyramid ABCDEM to the pyramid ADEM. [V. 18]
6
Similarly also it can be proved that, as the base FGHKL is to the base FGH, so is the pyramid FGHKLN to the pyramid FGHN.
6
And, since ADEM, FGHN are two pyramids which have triangular bases and equal height, therefore, as the base ADE is to the base FGH, so is the pyramid ADEM to the pyramid FGHN. [XII. 5]
6
But, as the base ADE is to the base ABCDE, so was the pyramid ADEM to the pyramid ABCDEM.
6
Therefore also, ex aequali, as the base ABCDE is to the base FGH, so is the pyramid ABCDEM to the pyramid FGHN. [V. 22]
6
But further, as the base FGH is to the base FGHKL, so also was the pyramid FGHN to the pyramid FGHKLN.
6
Therefore also, ex aequali, as the base ABCDE is to the base FGHKL, so is the pyramid ABCDEM to the pyramid FGHKLN. [V. 22] Q. E. D.
PROPOSITION 7.
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Any prism which has a triangular base is divided into three pyramids equal to one another which have triangular bases.
7
Let there be a prism in which the triangle ABC is the base and DEF its opposite; I say that the prism ABCDEF is divided into three pyramids equal to one another, which have triangular bases.
7
For let BD, EC, CD be joined.
