Proposition XIII.5 — If a straight line be cut in extreme and mean ratio, and the…

Proof

Extending by the greater segment

A C

A^{'}

(so

A^{'} A = A C

A B

the original), check that

A^{'} A : A B = A B : (A^{'} A + A B - A C)

, which reduces via the original extreme-and-mean relation to the same form.

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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…
XIII.1Proposition XIII.1If a straight line be cut in extreme and mean ratio, the square on the greater segment added to the half of the whole…
XIII.1Definition XIII.1A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment,…

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