PROPOSITION 18.
On a given straight line to describe a rectilineal figure similar and similarly situated to a given rectilineal figure.
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.
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]
Therefore the remaining angle
CFD is equal to the angle
AGB; [
I. 32]
therefore the triangle FCD is equiangular with the triangle GAB.
Therefore, proportionally, as
FD is to
GB, so is
FC to
GA, and
CD to
AB.
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]
Therefore the remaining angle at
E is equal to the remaining angle at
H; [
I. 32]
therefore the triangle FDE is equiangular with the triangle GBH; therefore, proportionally, as FD is to GB, so is FE to GH, and ED to HB. [VI. 4]
But it was also proved that, as
FD is to
GB, so is
FC to
GA, and
CD to
AB;
therefore also, as FC is to AG, so is CD to AB, and FE to GH, and further ED to HB.
And, since the angle
CFD is equal to the angle
AGB, and the angle
DFE to the angle
BGH, therefore the whole angle
CFE is equal to the whole angle
AGH.
For the same reason
the angle CDE is also equal to the angle ABH.
And the angle at
C is also equal to the angle at
A,
and the angle at E to the angle at H.
Therefore
AH is equiangular with
CE; and they have the sides about their equal angles proportional;
therefore the rectilineal figure AH is similar to the rectilineal figure CE. [VI. Def. 1]
Therefore on the given straight line
AB the rectilineal figure
AH has been described similar and similarly situated to the given rectilineal figure
CE. Q. E. F.
