Euclid Book 5 Proposition 12
Statement
If any number of ratios are equal (), then any one of these equal ratios is equal to the ratio of the sum of the antecedents to the sum of the consequents ().
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AB+EF+IJ
CD+GH+KL
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.]c.,EuclideanDistance[G.,H.]d.,EuclideanDistance[I.,J.]e.,EuclideanDistance[K.,L.]f.,,(EuclideanDistance[A.,B.]+EuclideanDistance[E.,F.]+EuclideanDistance[I.,J.])(EuclideanDistance[C.,D.]+EuclideanDistance[G.,H.]+EuclideanDistance[K.,L.]),
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EuclideanDistance[A.,B.]
EuclideanDistance[C.,D.]
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a.+c.+e.
b.+d.+f.
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Explanations
Let any number of magnitudes , , , , , be proportional, so that, as is to , so is to and to ; I say that, as is to , so are , , to , , .
For of, , let equimultiples , , be taken, and of , , other, chance, equimultiples , , . Then since, as is to , so is to , and to , and of , , equimultiples , , have been taken, and of , , other, chance, equimultiples , , , therefore, if is in excess of , is also in excess of , and of , if equal, equal, and if less, less; so that, in addition, if is in excess of , then , , are in excess of , , , if equal, equal, and if less, less.
Now and , , are equimultiples of and , , , since, if any number of magnitudes whatever are respectively equimultiples of any magnitudes equal in multitude, whatever multiple one of the magnitudes is of one, that multiple also will all be of all.
For the same reason and , , are also equimultiples of and , , ; therefore, as is to , so are , , to , , .
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For of
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Now
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For the same reason
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