Euclid Book 13 Proposition 5
Statement
If a line segment () is cut in the golden ratio (), and a segment () equal to the longer segment () is added to it, then the whole line segment () forms the golden ratio with the original ().
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Computational Explanation
GeometricScene{{A.,B.,C.,D.},{a.,b.}},Line[{A.,C.,B.}],EuclideanDistance[A.,C.]a.,EuclideanDistance[C.,B.]b.,,Line[{D.,A.,C.}],EuclideanDistance[D.,A.]EuclideanDistance[A.,C.],,
EuclideanDistance[A.,B.]
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EuclideanDistance[D.,B.] EuclideanDistance[A.,B.] EuclideanDistance[A.,B.] EuclideanDistance[A.,D.] |
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Explanations
For let the straight line be cut in extreme and mean ratio at the point , let be the greater segment, and let be equal to .
I say that the straight line has been cut in extreme and mean ratio at , and the original straight line is the greater segment.
For let the square be described on , and let the figure be drawn.
Since has been cut in extreme and mean ratio at , therefore the rectangle , is equal to the square on .
And is the rectangle , , and the square on ; therefore is equal to . But is equal to , and is equal to ; therefore is also equal to . Therefore the whole is equal to the whole .
And is the rectangle , , for is equal to ; and is the square on ; therefore the rectangle , is equal to the square on .
Therefore, as is to , so is to .
And is greater than ; therefore is also greater than .
Therefore has been cut in extreme and mean ratio at , and is the greater segment.
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I say that the straight line
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For let the square
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Since
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Therefore, as
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Therefore
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