---
book: 6
number: 13
id: "VI.13"
kind: "construction"
uses: ["[[book-3/proposition-31]]", "[[book-6/proposition-8]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:6.prop.13"
license: "CC-BY-SA-4.0"
---

# VI.13

*To two given straight lines to find a mean proportional*.

## Proof

Let *AB*, *BC* be the two given straight lines; thus it is required to find a mean proportional to *AB*, *BC*.

Let them be placed in a straight line, and let the semicircle *ADC* be described on *AC*; let *BD* be drawn from the point *B* at right angles to the straight line *AC*, and let *AD*, *DC* be joined.

Since the angle *ADC* is an angle in a semicircle, it is right. [[book-3/proposition-31|III. 31]]

And, since, in the right-angled triangle *ADC*, *DB* has been drawn from the right angle perpendicular to the base, therefore *DB* is a mean proportional between the segments of the base, *AB*, *BC*. [[book-6/proposition-8|VI. 8, Por.]]

Therefore to the two given straight lines *AB*, *BC* a mean proportional *DB* has been found. Q. E. F.