If four straight lines are proportional, then the rectangle contained by the extremes equals the rectangle contained by the means; and, if the rectangle contained by the extremes equals the rectangle contained by the means, then the four straight lines are proportional. | |
Let the four straight lines AB, CD, E, and F be proportional, so that AB is to CD as E is to F.
I say that the rectangle AB by F equals the rectangle CD by E. | |
Draw AG and CH from the points A and C at right angles to the straight lines AB and CD, and make AG equal to F, and CH equal to E. | I.11
I.3 |
Complete the parallelograms BG and DH. | I.31 |
Then since AB is to CD as E is to F, while E equals CH, and F equals AG, therefore AB is to CD as CH is to AG. | V.7 |
Therefore in the parallelograms BG and DH the sides about the equal angles are reciprocally proportional. | |
But those equiangular parallelograms in which the sides about the equal angles are reciprocally proportional are equal, therefore the parallelogram BG equals the parallelogram DH. | VI.14 |
And BG is the rectangle AB by F, for AG equals F, and DH is the rectangle CD by E, for E equals CH, therefore the rectangle AB by F equals the rectangle CD by E. | |
Next, let the rectangle AB by F be equal to the rectangle CD by E. I say that the four straight lines are proportional, so that AB is to CD as E is to F. | |
With the same construction, since the rectangle AB by F equals the rectangle CD by E, and the rectangle AB by F is BG, for AG equals F, and the rectangle CD by E is DH, for CH equals E, therefore BG equals DH. | |
And they are equiangular. But in equal and equiangular parallelograms the sides about the equal angles are reciprocally proportional. | VI.14 |
Therefore AB is to CD as CH is to AG. | V.7 |
But CH equals E, and AG to F, therefore AB is to CD as E is to F. | |
Therefore, if four straight lines are proportional, then the rectangle contained by the extremes equals the rectangle contained by the means; and, if the rectangle contained by the extremes equals the rectangle contained by the means, then the four straight lines are proportional. | |
Q.E.D. |
Book VI Introduction - Proposition VI.15 - Proposition VI.17.