Proposition 40
Theorem
Two triangles (△ABC , △DEF ) with the same area on equal bases (ￜBC ￜ = ￜEF ￜ ) which form parts of the same line, and on the same side of the line, are between the same parallels (infinite line AD ‖ infinite line BE ).
Commentary
1. Let △ABC and △DEF be two triangles with equal areas. Let them both be on the same side of equal bases (BC and EF ) that lie on the same line.
2. Then these two triangles are between the same parallels, infinite lineAD and infinite line BE .
3. This proposition is a generalization of the previous proposition, Book 1 Proposition 39, and is the converse of Book 1 Proposition 38.
2. Then these two triangles are between the same parallels, infinite line
3. This proposition is a generalization of the previous proposition, Book 1 Proposition 39, and is the converse of Book 1 Proposition 38.
Original statement
τὰ ἴσα τρίγωνα τὰ ἐπὶ ἴσων βάσϵων ὄντα καὶ ἐπὶ τὰ αὐτὰ μέρη καὶ ἐν ταῖς αὐταῖς παραλλήλοις ἐστίν.
English translation
Equal triangles which are on equal bases and on the same side are also in the same parallels.
