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 line AD 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.

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.

Under Development

Last updated on: 2022-04-14.

Proposition 40

Theorem

Commentary

Original statement

English translation

Computable version

Additional instances

Dependency graphs