Euclid Book 5 Proposition 7
Statement
The ratios of two equal magnitudes (ABCD) to a third are equal (). Similarly, the ratios of a magnitude to two equal magnitudes are equal ().
AB
EF
CD
EF
EF
AB
EF
CD
Computational Explanation
GeometricScene{A.,B.,C.,D.,E.,F.},{x.,y.,z.},Style[Line[{{A.,B.},{C.,D.}}],Red],Style[Line[{E.,F.}],Blue],EuclideanDistance[A.,B.]x.,EuclideanDistance[C.,D.]y.,x.y.,EuclideanDistance[E.,F.]z.,,,,
EuclideanDistance[A.,B.]
EuclideanDistance[E.,F.]
EuclideanDistance[C.,D.]
EuclideanDistance[E.,F.]
x.
z.
y.
z.
EuclideanDistance[E.,F.]
EuclideanDistance[A.,B.]
EuclideanDistance[E.,F.]
EuclideanDistance[C.,D.]
z.
x.
z.
y.
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Explanations
Let , be equal magnitudes and any other, chance, magnitude; I say that each of the magnitudes , has the same ratio to , and has the same ratio to each of the magnitudes , .
For let equimultiples, of , be taken, and of another, chance, multiple .
Then, since is the same multiple of that is of , while is equal to , therefore is equal to .
But is another, chance, magnitude.
If therefore is in excess of , is also in excess of , if equal to it, equal; and, if less, less.
And, are equimultiples of , , while is another, chance, multiple of ; therefore, as is to , so is to .
I say next that also has the same ratio to each of the magnitudes , .
For, with the same construction, we can prove similarly that is equal to ; and is some other magnitude.
If therefore is in excess of , it is also in excess of , if equal, equal; and, if less, less.
And is a multiple of , while , are other, chance, equimultiples of , ; therefore, as is to , so is to .
A
B
C
A
B
C
C
A
B
For let equimultiples
D
E
A
B
C
F
Then, since
D
A
E
B
A
B
D
E
But
F
If therefore
D
F
E
F
And
D
E
A
B
F
C
A
C
B
C
I say next that
C
A
B
For, with the same construction, we can prove similarly that
D
E
F
If therefore
F
D
E
And
F
C
D
E
A
B
C
A
C
B