Proposition XIII.1 — If a straight line be cut in extreme and mean ratio, the squ…

Proof

Let

A B

be cut at

C

in extreme and mean ratio with

A C > C B

. Let

D

be the midpoint of

A B

. Apply II.6:

(A B /2 + A C)^{2} = (A B /2)^{2} + A B \cdot A C + A C^{2}

. By the defining relation

A C^{2} = A B \cdot C B

, simplification gives

(A B /2 + A C)^{2} = 5 (A B /2)^{2}

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Depends on (3)

II.6Proposition II.6If a straight line be bisected and a straight line be added to it in a straight line, the rectangle contained by the…
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.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,…

Required by (dependents) (5)

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