To find a third proportional to two given straight lines.

Let *AB* and *AC* be the two given straight lines, and let them be placed so as to contain any angle.

It is required to find a third proportional to *AB* and *AC.*

Produce them to the points *D* and *E,* and make *BD* equal to *AC.* Join *BC,* and draw *DE* through *D* parallel to it.

Then since *BC* is parallel to a side *DE* of the triangle *ADE,* therefore, proportionally, *AB* is to *BD* as *AC* is to *CE.*

But *BD* equals *AC,* therefore *AB* is to *AC* as *AC* is to *CE.*

Therefore a third proportional *CE* has been found to two given straight lines *AB* and *AC.*

Q.E.F.

This construction is used in propositions VI.19, VI.22, and a few propositions in Book X.