Kitap 6
27
Of all the parallelograms applied to the same straight line and deficient by parallelogrammic figures similar and similarly situated to that described on the half of the straight line, that parallelogram is greatest which is applied to the half of the straight line and is similar to the defect.
27
Let AB be a straight line and let it be bisected at C; let there be applied to the straight line AB the parallelogram AD deficient by the parallelogrammic figure DB described on the half of AB, that is, CB; I say that, of all the parallelograms applied to AB and deficient by parallelogrammic figures similar and similarly situated to DB, AD is greatest.
27
For let there be applied to the straight line AB the parallelogram AF deficient by the parallelogrammic figure FB similar and similarly situated to DB; I say that AD is greater than AF.
27
For, since the parallelogram DB is similar to the parallelogram FB, they are about the same diameter. [VI. 26]
27
Let their diameter DB be drawn, and let the figure be described.
27
Then, since CF is equal to FE, [I. 43] and FB is common, therefore the whole CH is equal to the whole KE.
27
But CH is equal to CG, since AC is also equal to CB. [I. 36]
27
Therefore GC is also equal to EK.
27
Let CF be added to each; therefore the whole AF is equal to the gnomon LMN; so that the parallelogram DB, that is, AD, is greater than the parallelogram AF.
27
Therefore etc.
PROPOSITION 28.
28
To a given straight line to apply a parallelogram equal to a given rectilineal figure and deficient by a parallelogrammic figure similar to a given one : thus the given rectilineal figure must not be greater than the parallelogram described on the half of the straight line and similar to the defect.
