Source — Euclid's Elements, encoded as an rrxiv paper

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XIII.1 Proposition XIII.1

If a straight line be cut in extreme and mean ratio, the square on the greater segment added to the half of the whole is five times the square on the half.

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}

.

lines 74–74 in main.tex