equal to the Bafe DB; and the Triangle A B C is equal to the Triangle B C D. Wherefore, the Diameter B C bifects the Porallelogram ACDB; which was to be demonftrated.

PROPOSITION XXXV.

THEORE M.

Parallelograms conftituted upon the fame Bafe, and between the fame Parallels, are equal between themselves.

L ET ABCD, EBCF, be Parallelograms conftituted upon the fame Bafe BC, and between the fame Parallels A F and BC. I fay, the Paral lelogram ABCD is equal to the Parallelogram. EBCF.

For, becaufe A B C D is a Parallelogram, AD is

*

29 of this.

* equal to B C; and for the fame Reafon E F is equal * 34 of this. to BC; wherefore A D fhall be + equal to E F; but Ax. 1. DE is common. Therefore the whole A E is ‡ equal ‡ Ax. 2. to the whole D F. But A B is equal to DC; wherefore E A, A B, the two Sides of the Triangle A BE, are equal to the two Sides F D, DC, each to each; and the Angle F DC equal to the Angle E A B, the outward one to the inward one. Therefore the Base EB is equal to the Bafe CF, and the Triangle EAB † 4 of this. to the Triangle FD C. If the common Triangle DGE be taken from both, there will remain the Ax. 3. Trapezium ABG D, equal to the Trapezium FCGE; and if the Triangle G B C, which is common, be added, the Parallelogram A B C D will be equal to the Parallelogram E B CF. Therefore, Parallelograms conftituted upon the fame. Bafe, and between the fame Parallels, are equal between themselves; which was to be demonstrated.

Hyp.

PROPOSITION XXXVI.

THEOREM.

Parallelograms conftituted upon equal Bases, and between the fame Parallels, are equal between ' themselves.

LET the Parallelograms ABCD, EFGH, be conftituted upon the equal Bases B C, F G, and between the fame Parallels A H, BG. I fay, the Parallelogram A B C D is equal to the Parallelogram EFGH.

For join B E, CH. Then becaufe BC* is equal to F G, and F G to EH; BC will be likewife equal to EH; and they are parallel, and B E, C H, join them. But two Right Lines joining Right Lines, which are equal and parallel, towards the fame Parts, +33 of this are + equal and parallel: Wherefore EBCH is a 135 of this. Parallelogram, and is equal to the Parallelogram ABCD; for it has the fame Base B C, and is conftituted between the fame Parallels B C, A H. For the fame Reason, the Parallelogram E F G H is equal to the fame Parallelogram E BCH. Therefore the Parallelogram A B CD fhall be equal to the Parallelogram EF GH. And fo Parallelograms conftituted upon equal Bafes, and between the fame Parallels, are equal between themselves; which was to be demonftrated."

PROPOSITION XXXVII.

THEOREM.

Triangles conftituted upon the fame Base, and between the fame Parallels, are equal between themselves.

L

E T the Triangles A B C, D BC, be conftituted upon the fame Bafe B C, and between the fame Parallels A D, B C. I fay, the Triangle A B C is equal to the Triangle D BC.

For produce A D both Ways to the Points E and 31 of this. F; and through B draw* BE_parallel to C A; and through C, CF, parallel to B D.

Where .

*

Wherefore both EB CA, DBCF, are Parallelograms; and the Parallelogram EBC A is equal to * 35 of this. the Parallelogram DBCF; for they stand upon the fame Base BC, and between the fame Parallels B C, EF. But the Triangle ABC is + one half of the Paral·† 34 of this. lelogram, EBC A, because the Diameter A B bifects it; and the Triangle D B C is one half of the Parallelogram DBCF, for the Diameter DC bifects it. But Things that are the Halves of equal Things, are ‡ ‡ Ax. 7. equal between themfelves. Therefore the Triangle ABC is equal to the Triangle DBC. Wherefore, Triangles conftituted upon the fame Bafe, and between the fame Parallels, are equal between themfelves; which was to be demonstrated.

PROPOSITION XXXVIII.

THEOREM.

Triangles conftituted upon equal Bases, and between the fame Parallels, are equal between themselves.

LE

ET the Triangles A B C, DCE, be conftituted upon the equal Bafes BC, CE, and between the fame Parallels BE, A D. I fay, the Triangle ABC is equal to the Triangle DCE.

For, produce AD both Ways to the Points, G,

H; through B draw BG parallel to CA; and 31 of this. through E, E H, parallel to DC.

Wherefore both GBCA, DCEH, are Paralle

lograms; and the Parallelogram GBCA is + equal † 36 of this. to the Parallelogram DCEH: For they ftand upon equal Bases, B Č, C E, and between the fame Parallels BE, G H. But the Triangle ABC is ‡ one half 34 of this. of the Parallelogram GBCA, for the Diameter A B bifects it; and the Triangle DCE is one half of the Parallelogram DCEH, for the Diameter D E bifects it. But Things that are the Halves of equal Things, are equal between themfelves. Therefore * A*. 7. the Triangle A B C is equal to the Triangle DC E. Wherefore, Triangles conftituted upon equal Bafes, and between the fame Parallels, are equal between themfelves; which was to be demonftrated.

*