|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.||I.3
|Then since BC is parallel to a side DE of the triangle ADE, therefore, proportionally, AB is to BD as AC is to CE.||VI.2|
|But BD equals AC, therefore AB is to AC as AC is to CE.||V.7|
|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.
Next proposition: VI.12