Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/199

Rh and since it has been shewn that if the base HC be greater than the base CL, the triangle AHC is greater than the triangle ACL ; and if equal, equal ; and if less, less ; therefore as the base BC is to the base CD, so is the triangle ABC to the triangle ACD. [V. Definition 5.

And, because the parallelogram CE is double of the triangle ABC, and the parallelogram CF is double of the triangle ACD; [I.41. and that magnitudes have the same ratio which their equi- multiples have ; [V. 15. therefore the parallelogram EC is to the parallelogram CF as the triangle ABC is to the triangle ACD. But it has been shewn that the triangle ABC is to the triangle ACD as the base BC is to the base CD ; therefore the parallelogram EC is to the parallelogram CF as the base BC is to the base CD. [V. 11.

Wherefore, triangles &c.

From this it is plain that triangles and parallelograms which have equal altitudes, are to one an- other as their bases.

For, let the figures be placed so as to have their bases in the same straight line, and to be on the same side of it ; and having drawn perpendiculars from the vertices of the triangles to the bases, the straight line which joins the ver- tices is parallel to that in which their bases are ; [I. 33. because the perpendiculars are both equal and parallel to one another. [I. 28.

Then, if the same construction be made as in the pro- position, the demonstration will be the same.

PROPOSITION 2. THEOREM.

If a straight line he drawn parallel to one of the sides of a triangle, it shall cut the other sides, or those sides  produced, proportionally ; and if the sides, or the sides produced, be cut proportionally, the straight line which joins the points of section, shall be parallel to the re-maining side of the triangle.