Proof
Apply XII.4 in the limit of XII.3 iterations; the prism-sums
exhaust the pyramids (X.1), so the base-ratio is the pyramid-ratio.
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Depends on (3)
- X.1Proposition X.1Two unequal magnitudes being set out, if from the greater there be subtracted a magnitude greater than its half, and…
- XII.3Proposition XII.3Any pyramid which has a triangular base is divided into two pyramids equal and similar to one another, similar to the…
- XII.4Proposition XII.4If there be two pyramids of the same height which have triangular bases, and each of them be divided into two pyramids…
Required by (dependents) (4)
- XII.6Proposition XII.6Pyramids which are of the same height and have polygonal bases are to one another as the bases.
- XII.7Proposition XII.7Any prism which has a triangular base is divided into three pyramids equal to one another which have triangular bases.
- XII.8Proposition XII.8Similar pyramids which have triangular bases are in the triplicate ratio of their corresponding sides.
- XII.9Proposition XII.9In equal pyramids which have triangular bases the bases are reciprocally proportional to the heights; and those…
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