Book 6
28
Therefore GQ is also similar to GB; [VI. 21] therefore GQ is about the same diameter with GB. [VI. 26]
28
Let GQB be their diameter, and let the figure be described.
28
Then, since BG is equal to C, KM, and in them GQ is equal to KM, therefore the remainder, the gnomon UWV, is equal to the remainder C.
28
And, since PR is equal to OS, let QB be added to each; therefore the whole PB is equal to the whole OB.
28
But OB is equal to TE, since the side AE is also equal to the side EB; [I. 36] therefore TE is also equal to PB.
28
Let OS be added to each; therefore the whole TS is equal to the whole, the gnomon VWU.
28
But the gnomon VWU was proved equal to C; therefore TS is also equal to C.
28
Therefore to the given straight line AB there has been applied the parallelogram ST equal to the given rectilineal figure C and deficient by a parallelogrammic figure QB which is similar to D. Q. E. F.
PROPOSITION 29.
29
To a given straight line to apply a parallelogram equal to a given rectilineal figure and exceeding by a parallelogrammic figure similar to a given one.
29
Let AB be the given straight line, C the given rectilineal figure to which the figure to be applied to AB is required to be equal, and D that to which the excess is required to be similar; thus it is required to apply to the straight line AB a parallelogram equal to the rectilineal figure C and exceeding by a parallelogrammic figure similar to D.
