Euclid’s Elements

Book VI

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.

I.11

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.

III.31

Since the angle ADC is an angle in a semicircle, it is right.

VI.8,Cor

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.

Therefore a mean proportional BD has been found to the two given straight lines AB and BC.

Q.E.F.

Guide

This construction of the mean proportional was used before in II.14 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. Thus, x is the square root of ab. This mean proportional between a and b is also called the geometric mean of a and b.

When b is taken to have unit length, this construction gives the construction for the square root of a.

Use of this proposition

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

Referenced by

Book VIVI.25
Book XX.27, X.28