PROPOSITION 22.
If four straight lines be proportional,
the rectilineal figures similar and similarly described upon them will also be proportional;
and,
if the rectilineal figures similar and similarly described upon them be proportional,
the straight lines will themselves also be proportional.
Let the four straight lines
AB,
CD,
EF,
GH be proportional, so that, as
AB is to
CD, so is
EF to
GH, and let there be described on
AB,
CD the similar and similarly situated rectilineal figures
KAB,
LCD, and on
EF,
GH the similar and similarly situated rectilineal figures
MF,
NH; I say that, as
KAB is to
LCD, so is
MF to
NH.
For let there be taken a third proportional
O to
AB,
CD, and a third proportional
P to
EF,
GH. [
VI. 11]
Then since, as
AB is to
CD, so is
EF to
GH,
and, as CD is to O, so is GH to P, therefore,
ex aequali, as
AB is to
O, so is
EF to
P. [
V. 22]
But, as
AB is to
O, so is
KAB to
LCD, [
VI. 19, Por.]
and, as EF is to P, so is MF to NH; therefore also, as
KAB is to
LCD, so is
MF to
NH. [
V. 11]
Next, let
MF be to
NH as
KAB is to
LCD; I say also that, as
AB is to
CD, so is
EF to
GH.
For, if
EF is not to
GH as
AB to
CD,
let EF be to QR as AB to CD, [VI. 12] and on
QR let the rectilineal figure
SR be described similar and similarly situated to either of the two
MF,
NH. [
VI. 18]
Since then, as
AB is to
CD, so is
EF to
QR, and there have been described on
AB,
CD the similar and similarly situated figures
KAB,
LCD, and on
EF,
QR the similar and similarly situated figures
MF,
SR, therefore, as
KAB is to
LCD, so is
MF to
SR.
But also, by hypothesis,
as KAB is to LCD, so is MF to NH; therefore also, as MF is to SR, so is MF to NH. [V. 11]
Therefore
MF has the same ratio to each of the figures
NH,
SR;
therefore NH is equal to SR. [V. 9]
But it is also similar and similarly situated to it;
therefore GH is equal to QR.
And, since, as
AB is to
CD, so is
EF to
QR, while
QR is equal to
GH, therefore, as
AB is to
CD, so is
EF to
GH.
Therefore etc. Q. E. D.
