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. q.e.d.
Corollary. 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.
