## Proposition 17

 If three straight lines are proportional, then the rectangle contained by the extremes equals the square on the mean; and, if the rectangle contained by the extremes equals the square on the mean, then the three straight lines are proportional. Let the three straight lines A and B and C be proportional, so that A is to B as B is to C. I say that the rectangle A by C equals the square on B. Make D equal to B. I.3 Then, since A is to B as B is to C, and B equals D, therefore A is to B as D is to C. V.7 V.11 But, if four straight lines are proportional, then the rectangle contained by the extremes equals the rectangle contained by the means. VI.16 Therefore the rectangle A by C equals the rectangle B by D. But the rectangle B by D is the square on B, for B equals D, therefore the rectangle A by C equals the square on B. Next, let the rectangle A by C equal the square on B. I say that A is to B as B is to C. With the same construction, since the rectangle A by C equals the square on B, while the square on B is the rectangle B by D, for B equals D, therefore the rectangle A by C equals the rectangle B by D. But, if the rectangle contained by the extremes equals that contained by the means, then the four straight lines are proportional. VI.16 Therefore A is to B as D is to C. But B equals D, therefore A is to B as B is to C. Therefore, if three straight lines are proportional, then the rectangle contained by the extremes equals the square on the mean; and, if the rectangle contained by the extremes equals the square on the mean, then the three straight lines are proportional. Q.E.D.
This is obviously a special case of the previous proposition. It is used very frequently in Books X and XIII.

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