Kitap 12
12
And since, owing to the similarity of the triangles LKB, NMF, as LB is to BK, so is NF to FM, and, owing to the similarity of the triangles BKT, FMP, as KB is to BT, so is MF to FP, therefore, ex aequali, as LB is to BT, so is NF to FP. [V. 22]
12
Again since, owing to the similarity of the triangles LTK, NPM, as LT is to TK, so is NP to PM, and, owing to the similarity of the triangles TKB, PMF, as KT is to TB, so is MP to PF; therefore, ex aequali, as LT is to TB, so is NP to PF. [V. 22]
12
But it was also proved that, as TB is to BL, so is PF to FN.
12
Therefore, ex aequali, as TL is to LB, so is PN to NF. [V. 22]
12
Therefore in the triangles LTB, NPF the sides are proportional; therefore the triangles LTB, NPF are equiangular; [VI. 5] hence they are also similar. [VI. Def. I]
12
Therefore the pyramid of which the triangle BKT is the base and the point L the vertex is also similar to the pyramid of which the triangle FMP is the base and the point N the vertex, for they are contained by similar planes equal in multitude. [XI. Def. 9]
12
But similar pyramids which have triangular bases are to one another in the triplicate ratio of their corresponding sides. [XII. 8]
12
Therefore the pyramid BKTL has to the pyramid FMPN the ratio triplicate of that which BK has to FM.
12
Similarly, by joining straight lines from A, W, D, V, C, U to K, and from E, S, H, R, G, Q to M, and setting up on each of the triangles pyramids which have the same vertex with the cones, we can prove that each of the similarly arranged pyramids will also have to each similarly arranged pyramid the ratio triplicate of that which the corresponding side BK has to the corresponding side FM, that is, which BD has to FH.
