Kitap 12
3
Therefore the angle EHG is equal to the angle KDL.
3
And, since the two straight lines EH, HG are equal to the two KD, DL respectively, and the angle EHG is equal to the angle KDL, therefore the base EG is equal to the base KL; [I. 4] therefore the triangle EHG is equal and similar to the triangle KDL.
3
For the same reason the triangle AEG is also equal and similar to the triangle HKL.
3
Therefore the pyramid of which the triangle AEG is the base and the point H the vertex is equal and similar to the pyramid of which the triangle HKL is the base and the point D the vertex. [XI. Def. 10]
3
And, since HK has been drawn parallel to AB, one of the sides of the triangle ADB, the triangle ADB is equiangular to the triangle DHK, [I. 29] and they have their sides proportional; therefore the triangle ADB is similar to the triangle DHK. [VI. Def. 1]
3
For the same reason the triangle DBC is also similar to the triangle DKL, and the triangle ADC to the triangle DLH.
3
Now, since the two straight lines BA, AC meeting one another are parallel to the two straight lines KH, HL meeting one another, not in the same plane, they will contain equal angles. [XI. 10]
3
Therefore the angle BAC is equal to the angle KHL.
3
And, as BA is to AC, so is KH to HL; therefore the triangle ABC is similar to the triangle HKL.
3
Therefore also the pyramid of which the triangle ABC is the base and the point D the vertex is similar to the pyramid of which the triangle HKL is the base and the point D the vertex.
