## 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. Thus, x is the square root of ab.

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.

Next proposition: VI.14

Previous: VI.12

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