Proposition 38
Theorem
Two triangles (△ABC , △DEF ) on equal bases (ￜBC ￜ = ￜEF ￜ ) and between the same parallels (infinite line AD ‖ infinite line BC ) have the same area.
Commentary
1. Let △ABC and △DEF be two triangles that are on equal bases (BC and EF ) and between the same parallel lines (infinite line AD and infinite line BE ).
2. Then these two triangles are equal in area.
3. This proposition is a generalization of the previous proposition, Book 1 Proposition 37, and is the converse of Book 1 Proposition 40.
2. Then these two triangles are equal in area.
3. This proposition is a generalization of the previous proposition, Book 1 Proposition 37, and is the converse of Book 1 Proposition 40.
Original statement
τὰ τρίγωνα τὰ ἐπὶ ἴσων βάσϵων ὄντα καὶ ἐν ταῖς αὐταῖς παραλλήλοις ἴσα ἀλλήλοις ἐστίν.
English translation
Triangles which are on equal bases and in the same parallels are equal to one another.
