| 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
I.31 | |
| 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. | ||
| Q.E.F. | ||
Book VI Introduction - Proposition VI.10 - Proposition VI.12.