Euclid Book 5 Proposition 25
Statement
If the greatest of four magnitudes is to the second as the third is to the least (), then the sum of the greatest and the least is greater than the sum of the remaining two (AB+GH>CD+EF).
AB
CD
EF
GH
Computational Explanation
GeometricScene{{A.,B.,C.,D.,E.,F.,G.,H.},{a.,b.,c.,d.}},Style[Line[{A.,B.}],Red],Style[Line[{C.,D.}],Pink],Style[Line[{E.,F.}],Blue],Style[Line[{G.,H.}],Cyan],EuclideanDistance[A.,B.]a.,EuclideanDistance[C.,D.]b.,EuclideanDistance[E.,F.]c.,EuclideanDistance[G.,H.]d.,a.>b.>c.>d.,,{EuclideanDistance[A.,B.]+EuclideanDistance[G.,H.]>EuclideanDistance[C.,D.]+EuclideanDistance[E.,F.],a.+d.>b.+c.}
a.
b.
c.
d.
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Explanations
Let the four magnitudes , , , be proportional so that, as is to , so is to , and let be the greatest of them and the least; I say that , are greater than , .
For let be made equal to , and equal to .
Since, as is to , so is to , and is equal to , and to , therefore, as is to , so is to .
And since, as the whole is to the whole , so is the part subtracted to the part subtracted, the remainder will also be to the remainder as the whole is to the whole .
But is greater than ; therefore is also greater than .
And, since is equal to , and to , therefore , are equal to , .
And if,, being unequal, and greater, , be added to and , be added to , it follows that , are greater than , .
AB
CD
E
F
AB
CD
E
F
AB
F
AB
F
CD
E
For let
AG
E
CH
F
Since, as
AB
CD
E
F
E
AG
F
CH
AB
CD
AG
CH
And since, as the whole
AB
CD
AG
CH
GB
HD
AB
CD
But
AB
CD
GB
HD
And, since
AG
E
CH
F
AG
F
CH
E
And if,
GB
HD
GB
AG
F
GB
CH
E
HD
AB
F
CD
E