Proposition VI.30 — To cut a given finite straight line in extreme and mean rati…

Proof

Apply to the given line a parallelogram equal to the square on it, exceeding by a square (VI.29). The exceeding square's side

x

satisfies

x (a + x) = a^{2}

, equivalently

a : x = x : (a - x)

in the language of II.11 – the golden section.

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Full neighborhood

Depends on (3)

II.11Proposition II.11To cut a given straight line so that the rectangle contained by the whole and one of the segments is equal to the…
VI.17Proposition VI.17If three straight lines be proportional, the rectangle contained by the extremes is equal to the square on the mean;…
VI.29Proposition VI.29To a given straight line to apply a parallelogram equal to a given rectilineal figure and exceeding by a…

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