Kitap 12
3
Any pyramid which has a triangular base is divided into two pyramids equal and similar to one another, similar to the whole and having triangular bases, and into two equal prisms; and the two prisms are greater than the half of the whole pyramid.
3
Let there be a pyramid of which the triangle ABC is the base and the point D the vertex; I say that the pyramid ABCD is divided into two pyramids equal to one another, having triangular bases and similar to the whole pyramid, and into two equal prisms; and the two prisms are greater than the half of the whole pyramid.
3
For let AB, BC, CA, AD, DB, DC be bisected at the points E, F, G, H, K, L, and let HE, EG, GH, HK, KL, LH, KF, FG be joined.
3
Since AE is equal to EB, and AH to DH, therefore EH is parallel to DB. [VI. 2]
3
For the same reason HK is also parallel to AB.
3
Therefore HEBK is a parallelogram; therefore HK is equal to EB. [I. 34]
3
But EB is equal to EA; therefore AE is also equal to HK.
3
But AH is also equal to HD; therefore the two sides EA, AH are equal to the two sides KH, HD respectively, and the angle EAH is equal to the angle KHD; therefore the base EH is equal to the base KD. [I. 4]
3
Therefore the triangle AEH is equal and similar to the triangle HKD.
3
For the same reason the triangle AHG is also equal and similar to the triangle HLD.
3
Now, since two straight lines EH, HG meeting one another are parallel to two straight lines KD, DL meeting one another, and are not in the same plane, they will contain equal angles. [XI. 10]
