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Proposition 12

To find a fourth proportional to three given straight lines.
Let A and B and C be the three given straight lines. It is required to find a fourth proportional to A, B, and C.
| Set out two straight lines DE and DF containing any angle EDF. Make DG equal to A, GE equal to B, and DH equal to C. Join GH, and draw EF through E parallel to it. | I.3 I.31
Then since GH is parallel to a side EF of the triangle DEF, therefore DG is to GE as DH is to HF. | VI.2
But DG equals A and GE to B, and DH to C, therefore A is to B as C is to HF. | V.7
Therefore a fourth proportional HF has been found to the three given straight lines A, B, and C.
Q.E.F.

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Of course, the previous proposition is a special case of this one.

This proposition is used in the proofs of VI.22, VI.23, and half a dozen propositions in Book X.

Book VI Introduction - Proposition VI.11 - Proposition VI.13.

© 1996
D.E.Joyce
Clark University