| 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. | ||
Book VI Introduction - Proposition VI.16 - Proposition VI.18.