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
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
be bisected at E
; let there be described on EB
the parallelogram BF
similar and similarly situated to D
; and let GH
be constructed at once equal to the sum of BF
and similar and similarly situated to D
. [VI. 25
correspond to FL
Now, since GH
is greater than FB
, therefore KH
is also greater than FL
, and KG
be produced, let FLM
be equal to KH
, and FEN
, and let MN
be completed; therefore MN is both equal and similar to GH.
is similar to EL
; therefore MN is also similar to EL; [VI. 21]
is about the same diameter with MN
. [VI. 26
Let their diameter FO
be drawn, and let the figure be described.
is equal to EL
, while GH
is equal to MN
, therefore MN
is also equal to EL
be subtracted from each; therefore the remainder, the gnomon XWV, is equal to C.
Now, since AE
is equal to EB
, AN is also equal to NB [I. 36], that is, to LP [I. 43].
be added to each; therefore the whole AO is equal to the gnomon VWX.
But the gnomon VWX
is equal to C
; therefore AO is also equal to C.
Therefore to the given straight line AB
there has been applied the parallelogram AO
equal to the given rectilineal figure C
and exceeding by a parallelogrammic figure QP
which is similar to D
, since PQ
is also similar to EL
]. Q. E. F.