The First Six Books of the Elements of Euclid

by Euclid; John Casey

8. If three magnitudes be continual proportionals, the first is to the third as the square of the difference between the first and second is to the square of the difference between the second and third.

9. If a line AB, cut harmonically in C and D, be bisected in O; prove OC, OB, OD are continual proportionals.

10. In the same case, if O′ be the middle point of CD; prove OO′² = OB² + O′D².

11. And AB(AC + AD) = 2AC.AD, or

+

=

.