Proposition 39
Theorem
Two triangles (△ABC , △DBC ) with the same area on the same side of the same base (BC ) are between the same parallels (infinite line AD ‖ infinite line BC ).
Commentary
1. Let △ABC and △DBC be two triangles with equal areas. Let them both be on the same base BC , and on the same side of BC .
2. Then these two triangles are between the same parallels, infinite lineAD and infinite line BC .
3. This proposition is a special case of the next proposition, Book 1 Proposition 40, and is the converse of Book 1 Proposition 37.
2. Then these two triangles are between the same parallels, infinite line
3. This proposition is a special case of the next proposition, Book 1 Proposition 40, and is the converse of Book 1 Proposition 37.
Original statement
τὰ ἴσα τρίγωνα τὰ ἐπὶ τῆς αὐτῆς βάσϵως ὄντα καὶ ἐπὶ τὰ αὐτὰ μέρη καὶ ἐν ταῖς αὐταῖς παραλλήλοις ἐστίν.
English translation
Equal triangles which are on the same base and on the same side are also in the same parallels.
