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

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To two given straight lines to find a mean proportional.

Lay the two segments end-to-end as

A B

,

B C

on a single line; on

A C

as diameter describe a semicircle (Postulate 3, III.31 implicit). Erect the perpendicular

B D

at

B

to the diameter; then

△ A B D

,

△ B D C

, and

△ A B C

are similar by VI.8, so

A B : B D = B D : B C

, making

B D

the required mean proportional.