Euclid's Elements, Book VI, Proposition 13

Proposition 13

To find a mean proportional to two given straight lines.
Let AB and BC be the two given straight lines. It is required to find a mean proportional to AB and BC.
	Place them in a straight line, and describe the semicircle ADC on AC. Draw BD from the point B at right angles to the straight line AC, and join AD and DC.	I.11
	Since the angle ADC is an angle in a semicircle, it is right.	III.31
And, since, in the right-angled triangle ADC, BD has been drawn from the right angle perpendicular to the base, therefore BD is a mean proportional between the segments of the base, AB and BC.		VI.8,Cor
Therefore a mean proportional BD has been found to the two given straight lines AB and BC.
Q.E.F.

This construction of the mean proportional was used before in II.4 to find a square equal to a given rectangle. By proposition VI.17 coming up, the two constructions are equivalent. That is the mean proportional between two lines is the side of a square equal to the rectangle contained by the two lines. Algebraically, a : x = x : b if and only if ab = x².

This construction is used in the proofs of propositions VI.25, X.27, and X.28.

Book VI Introduction - Proposition VI.12 - Proposition VI.14.