PROPOSITION 27.
To find medial straight lines commensurable in square only which contain a rational rectangle.
Let two rational straight lines
A,
B commensurable in square only be set out; let
C be taken a mean proportional between
A,
B, [
VI. 13] and let it be contrived that,
as A is to B, so is C to D. [VI. 12]
Then, since
A,
B are rational and commensurable in square only, the rectangle
A,
B, that is, the square on
C [
VI.17], is medial. [
X. 21]
Therefore
C is medial. [
X. 21]
And since, as
A is to
B, so is
C to
D, and
A,
B are commensurable in square only, therefore
C,
D are also commensurable in square only. [
X. 11]
And
C is medial; therefore
D is also medial. [
X. 23, addition]
Therefore
C,
D are medial and commensurable in square only.
I say that they also contain a rational rectangle.
For since, as
A is to
B, so is
C to
D, therefore, alternately, as
A is to
C, so is
B to
D. [
V. 16]
But, as
A is to
C, so is
C to
B; therefore also, as
C is to
B, so is
B to
D; therefore the rectangle
C,
D is equal to the square on
B.
But the square on
B is rational; therefore the rectangle
C,
D is also rational.
Therefore medial straight lines commensurable in square only have been found which contain a rational rectangle. Q. E. D.