PROPOSITION 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.
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.
Let
AB 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,
C and similar and similarly situated to
D. [
VI. 25]
Let
KH correspond to
FL and
KG to
FE.
Now, since
GH is greater than
FB, therefore
KH is also greater than
FL, and
KG than
FE.
Let
FL,
FE be produced, let
FLM be equal to
KH, and
FEN to
KG, and let
MN be completed;
therefore MN is both equal and similar to GH.
But
GH is similar to
EL;
therefore MN is also similar to EL; [VI. 21] therefore
EL is about the same diameter with
MN. [
VI. 26]
Let their diameter
FO be drawn, and let the figure be described.
Since
GH is equal to
EL,
C, while
GH is equal to
MN, therefore
MN is also equal to
EL,
C.
Let
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].
Let
EO 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 [
VI. 24]. Q. E. F.
