Euclid Book 5 Proposition 19
Statement
If a whole is to another whole as a subtracted part is to another subtracted part (), then the remainder of the former is also to the remainder of the latter as the former whole is to the latter ().
AB
CD
AE
CF
EB
FD
AB
CD
Computational Explanation
GeometricScene{{A.,B.,C.,D.,E.,F.},{a.,b.,c.,d.}},Line[{A.,E.,B.}],Line[{C.,F.,D.}],Style[Line[{A.,E.}],Red],Style[Line[{E.,B.}],Pink],Style[Line[{C.,F.}],Blue],Style[Line[{F.,D.}],Cyan],EuclideanDistance[A.,E.]a.,EuclideanDistance[E.,B.]b.,EuclideanDistance[C.,F.]c.,EuclideanDistance[F.,D.]d.,,,,
EuclideanDistance[A.,B.]
EuclideanDistance[C.,D.]
EuclideanDistance[A.,E.]
EuclideanDistance[C.,F.]
a.+b.
c.+d.
a.
c.
EuclideanDistance[E.,B.]
EuclideanDistance[F.,D.]
EuclideanDistance[A.,B.]
EuclideanDistance[C.,D.]
b.
d.
a.+b.
c.+d.
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Explanations
For since, as is to , so is to , alternately also, as is to , so is to .
And, since the magnitudes are proportional componendo, they will also be proportional separando, that is, as is to , so is to , and, alternately, as is to , so is to .
But, as is to , so by hypothesis is the whole to the whole .
Therefore also the remainder will be to the remainder as the whole is to the whole .
AB
CD
AE
CF
BA
AE
DC
CF
And, since the magnitudes are proportional componendo, they will also be proportional separando, that is, as
BE
EA
DF
CF
BE
DF
EA
FC
But, as
AE
CF
AB
CD
Therefore also the remainder
EB
FD
AB
CD