Euclid Book 5 Proposition 11
Statement
If two ratios are equal to a third (, ), then they are equal to each other ().
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Computational Explanation
GeometricScene{{A.,B.,C.,D.,E.,F.,G.,H.,I.,J.,K.,L.},{a.,b.,c.,d.,e.,f.}},Style[Line[{A.,B.}],Red],Style[Line[{C.,D.}],Red],Style[Line[{E.,F.}],Blue],Style[Line[{G.,H.}],Blue],Style[Line[{I.,J.}],Green],Style[Line[{K.,L.}],Green],EuclideanDistance[A.,B.]a.,EuclideanDistance[C.,D.]b.,EuclideanDistance[E.,F.]e.,EuclideanDistance[G.,H.]f.,EuclideanDistance[I.,J.]c.,EuclideanDistance[K.,L.]d.,,,,
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EuclideanDistance[A.,B.]
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Explanations
For, as is to , so let be to , and, as is to , so let be to ; I say that, as is to , so is to .
For of, , let equimultiples , , be taken, and of , , other, chance, equimultiples , , .
Then since, as is to , so is to , and of , equimultiples , have been taken, and of , other, chance, equimultiples , , therefore, if is in excess of , is also in excess of , if equal, equal, and if less, less.
Again, since, as is to , so is to , and of , equimultiples , have been taken, and of , other, chance, equimultiples , , therefore, if is in excess of , is also in excess of , if equal, equal, and if less, less.
But we saw that, if was in excess of , was also in excess of ; if equal, equal; and if less, less; so that, in addition, if is in excess of , is also in excess of , if equal, equal, and if less, less.
And, are equimultiples of , while , are other, chance, equimultiples of , ; therefore, as is to , so is to.
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For of
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Then since, as
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Again, since, as
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But we saw that, if
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And
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