Book 6
16
For, with the same construction, since the rectangle AB, F is equal to the rectangle CD, E, and the rectangle AB, F is BG, for AG is equal to F, and the rectangle CD, E is DH, for CH is equal to E, therefore BG is equal to DH.
16
And they are equiangular
16
But in equal and equiangular parallelograms the sides about the equal angles are reciprocally proportional. [VI. 14]
16
Therefore, as AB is to CD, so is CH to AG.
16
But CH is equal to E, and AG to F; therefore, as AB is to CD, so is E to F.
16
Therefore etc. Q. E. D.
PROPOSITION 17
17
If three straight lines be proportional, the rectangle contained by the extremes is equal to the square on the mean; and, if the rectangle contained by the extremes be equal to the square on the mean, the three straight lines will be proportional.
17
Let the three straight lines A, B, C be proportional, so that, as A is to B, so is B to C; I say that the rectangle contained by A, C is equal to the square on B.
17
Let D be made equal to B.
17
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.
17
But, if four straight lines be proportional, the rectangle contained by the extremes is equal to the rectangle contained by the means. [VI. 16]
17
Therefore the rectangle A, C is equal to the rectangle B, D.
17
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.
17
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.
17
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.
17
But, if the rectangle contained by the extremes be equal to that contained by the means, the four straight lines are proportional. [VI. 16]
17
Therefore, as A is to B, so is D to C.
17
But B is equal to D; therefore, as A is to B, so is B to C.
17
Therefore etc. Q. E. D.
PROPOSITION 18.
