Book 1
35
Let DGE be subtracted from each; therefore the trapezium ABGD which remains is equal to the trapezium EGCF which remains. [C.N. 3]
35
Let the triangle GBC be added to each; therefore the whole parallelogram ABCD is equal to the whole parallelogram EBCF. [C.N. 2]
35
Therefore etc.
35
Q. E. D.
Proposition 36.
36
Enunciation Parallelograms which are on equal bases and in the same parallels are equal to one another.
36
Proof. Let ABCD, EFGH be parallelograms which are on equal bases BC, FG and in the same parallels AH, BG; I say that the parallelogram ABCD is equal to EFGH.
36
For let BE, CH be joined.
36
Then, since BC is equal to FG while FG is equal to EH, BC is also equal to EH. [C.N. 1]
36
But they are also parallel.
36
And EB, HC join them; but straight lines joining equal and parallel straight lines (at the extremities which are) in the same directions (respectively) are equal and parallel. [I. 33]
36
Therefore EBCH is a parallelogram. [I. 34]
36
And it is equal to ABCD; for it has the same base BC with it, and is in the same parallels BC, AH with it. [I. 35]
36
For the same reason also EFGH is equal to the same EBCH; [I. 35] so that the parallelogram ABCD is also equal to EFGH. [C.N. 1]
36
Therefore etc. Q. E. D.
Proposition 37.
37
Enunciation Triangles which are on the same base and in the same parallels are equal to one another.
37
Proof. Let ABC, DBC be triangles on the same base BC and in the same parallels AD, BC; I say that the triangle ABC is equal to the triangle DBC.
37
Let AD be produced in both directions to E, F; through B let BE be drawn parallel to CA, [I. 31] and through C let CF be drawn parallel to BD. [I. 31]
37
Then each of the figures EBCA, DBCF is a parallelogram; and they are equal,
37
for they are on the same base BC and in the same parallels BC, EF. [I. 35]
37
Moreover the triangle ABC is half of the parallelogram EBCA; for the diameter AB bisects it. [I. 34]
37
And the triangle DBC is half of the parallelogram DBCF; for the diameter DC bisects it. [I. 34]
37
[But the halves of equal things are equal to one another.]
