elem.6.17
若三条线段A、B、C成比例，即A比B等于B比C，则A与C所成矩形等于B上的正方形；反之亦然。
三线 A、B、C 成连比 A:B=B:C，即 A·C = B²；作辅助线 D=B 后由命题 16 推得结论。
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Let D be made equal to B. Then, since, as A is to B, so is B to C, and B is equal to D, therefore, as A is to B, so is D to C. But, if four straight lines be proportional, the rectangle contained by the extremes is equal to the rectangle contained by the means.
设D等于B。由A比B等于B比C，且B等于D，得A比B等于D比C。
[VI. 16] Therefore the rectangle A, C is equal to the rectangle B, D. But the rectangle B, D is the square on B, for B is equal to D; therefore the rectangle contained by A, C is equal to the square on B.
根据VI.16，若四条线段成比例，则外项所成矩形等于内项所成矩形，故A与C所成矩形等于B与D所成矩形。
Next, let the rectangle A, C be equal to the square on B; I say that, as A is to B, so is B to C. For, with the same construction, since the rectangle A, C is equal to the square on B, while the square on B is the rectangle B, D, for B is equal to D, therefore the rectangle A, C is equal to the rectangle B, D. But, if the rectangle contained by the extremes be equal to that contained by the means, the four straight lines are proportional.
因B等于D，B与D所成矩形即B上的正方形，所以A与C所成矩形等于B上的正方形。
[VI. 16] Therefore, as A is to B, so is D to C. But B is equal to D; therefore, as A is to B, so is B to C.
反之，若A与C所成矩形等于B上的正方形，同理可得A比B等于B比C。