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