Kitap 6
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.
18
On a given straight line to describe a rectilineal figure similar and similarly situated to a given rectilineal figure.
18
Let AB be the given straight line and CE the given rectilineal figure; thus it is required to describe on the straight line AB a rectilineal figure similar and similarly situated to the rectilineal figure CE.
18
Let DF be joined, and on the straight line AB, and at the points A, B on it, let the angle GAB be constructed equal to the angle at C, and the angle ABG equal to the angle CDF. [I. 23]
18
Therefore the remaining angle CFD is equal to the angle AGB; [I. 32] therefore the triangle FCD is equiangular with the triangle GAB.
18
Therefore, proportionally, as FD is to GB, so is FC to GA, and CD to AB.
18
Again, on the straight line BG, and at the points B, G on it, let the angle BGH be constructed equal to the angle DFE, and the angle GBH equal to the angle FDE. [I. 23]
