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(diagram 2 = general diagram)
note on the theorem
(diagram 1) 7. Let again line AG be a balance, and let B be its middle, and let there be hung at B triangle GDH, and let triangle GDH be scalene having base DH, but height equal to half the balance,(*) and let triangle DGH be suspended from points B, G, and area Z suspended at A incline equally to triangle GDH, holding as it is now positioned. Similarly it will be shown that area Z is a third part of triangle GDH.
(diagram 2) For let some other area also be suspended from A that is a third part of triangle BGH. (diagram 3) Triangle BDG, in fact, will incline equally to ZL. And so since triangle BGH inclines equally to L, and BDG with ZL, and ZL is a third of BGD, it is clear that triangle GDH is also three-times Z.
(*) That is, BG is the height of HGD. The distance HD is unimportant.