Page 170 - Euclid's Elements of Geometry

ST EW þ.

ELEMENTS BOOK 6

BC about the equal angles are reciprocally proportional,
that is to say, that as DB is to BE, so GB (is) to BF.
Ε Γ E C

Ζ Η F G
Β B

Α ∆ A D
Συμπεπληρώσθω γὰρ τὸ ΖΕ παραλληλόγραμμον. ἐπεὶ For let the parallelogram FE have been completed.
οὖν ἴσον ἐστὶ τὸ ΑΒ παραλληλόγραμμον τῷ ΒΓ παραλλη- Therefore, since parallelogram AB is equal to parallelo-
λογράμμῳ, ἄλλο δέ τι τὸ ΖΕ, ἔστιν ἄρα ὡς τὸ ΑΒ πρὸς τὸ gram BC, and FE (is) some other (parallelogram), thus
ΖΕ, οὕτως τὸ ΒΓ πρὸς τὸ ΖΕ. ἀλλ᾿ ὡς μὲν τὸ ΑΒ πρὸς as (parallelogram) AB is to FE, so (parallelogram) BC
τὸ ΖΕ, οὕτως ἡ ΔΒ πρὸς τὴν ΒΕ, ὡς δὲ τὸ ΒΓ πρὸς τὸ (is) to FE [Prop. 5.7]. But, as (parallelogram) AB (is) to
ΖΕ, οὕτως ἡ ΗΒ πρὸς τὴν ΒΖ· καὶ ὡς ἄρα ἡ ΔΒ πρὸς τὴν FE, so DB (is) to BE, and as (parallelogram) BC (is) to
ΒΕ, οὕτως ἡ ΗΒ πρὸς τὴν ΒΖ. τῶν ἄρα ΑΒ, ΒΓ παραλ- FE, so GB (is) to BF [Prop. 6.1]. Thus, also, as DB (is)
ληλογράμμων ἀντιπεπόνθασιν αἱ πλευραὶ αἱ περὶ τὰς ἴσας to BE, so GB (is) to BF. Thus, in parallelograms AB
γωνίας. and BC the sides about the equal angles are reciprocally
᾿Αλλὰ δὴ ἔστω ὡς ἡ ΔΒ πρὸς τὴν ΒΕ, οὕτως ἡ ΗΒ πρὸς proportional.
τὴν ΒΖ· λέγω, ὅτι ἴσον ἐστὶ τὸ ΑΒ παραλληλόγραμμον τῷ And so, let DB be to BE, as GB (is) to BF. I say that
ΒΓ παραλληλογράμμῳ. parallelogram AB is equal to parallelogram BC.
᾿Επεὶ γάρ ἐστιν ὡς ἡ ΔΒ πρὸς τὴν ΒΕ, οὕτως ἡ ΗΒ For since as DB is to BE, so GB (is) to BF, but as
πρὸς τὴν ΒΖ, ἀλλ᾿ ὡς μὲν ἡ ΔΒ πρὸς τὴν ΒΕ, οὕτως τὸ DB (is) to BE, so parallelogram AB (is) to parallelo-
ΑΒ παραλληλόγραμμον πρὸς τὸ ΖΕ παραλληλόγραμμον, ὡς gram FE, and as GB (is) to BF, so parallelogram BC
δὲ ἡ ΗΒ πρὸς τὴν ΒΖ, οὕτως τὸ ΒΓ παραλληλόγραμμον (is) to parallelogram FE [Prop. 6.1], thus, also, as (par-
πρὸς τὸ ΖΕ παραλληλόγραμμον, καὶ ὡς ἄρα τὸ ΑΒ πρὸς allelogram) AB (is) to FE, so (parallelogram) BC (is)
ieþ
τὸ ΖΕ, οὕτως τὸ ΒΓ πρὸς τὸ ΖΕ· ἴσον ἄρα ἐστὶ τὸ ΑΒ to FE [Prop. 5.11]. Thus, parallelogram AB is equal to
παραλληλόγραμμον τῷ ΒΓ παραλληλογράμμῳ. parallelogram BC [Prop. 5.9].
Τῶν ἄρα ἴσων τε καὶ ἰσογωνίων παραλληλογράμμων Thus, in equal and equiangular parallelograms the
ἀντιπεπόνθασιν αἱ πλευραὶ αἱ περὶ τὰς ἴσας γωνίας· καὶ ὧν sides about the equal angles are reciprocally propor-
ἰσογωνίων παραλληλογράμμων ἀντιπεπόνθασιν αἱ πλευραὶ tional. And those equiangular parallelograms in which
αἱ περὶ τὰς ἴσας γωνίας, ἴσα ἐστὶν ἐκεῖνα· ὅπερ ἔδει δεῖξαι. the sides about the equal angles are reciprocally propor-
tional are equal. (Which is) the very thing it was required
to show.

.
Proposition 15
Τῶν ἴσων καὶ μίαν μιᾷ ἴσην ἐχόντων γωνίαν τριγώνων In equal triangles also having one angle equal to one
ἀντιπεπόνθασιν αἱ πλευραὶ αἱ περὶ τὰς ἴσας γωνίας· καὶ ὧν (angle) the sides about the equal angles are reciprocally
μίαν μιᾷ ἴσην ἐχόντων γωνίαν τριγώνων ἀντιπεπόνθασιν αἱ proportional. And those triangles having one angle equal
πλευραὶ αἱ περὶ τὰς ἴσας γωνίας, ἴσα ἐστὶν ἐκεῖνα. to one angle for which the sides about the equal angles
῎Εστω ἴσα τρίγωνα τὰ ΑΒΓ, ΑΔΕ μίαν μιᾷ ἴσην ἔχοντα (are) reciprocally proportional are equal.
γωνίαν τὴν ὑπὸ ΒΑΓ τῇ ὑπὸ ΔΑΕ· λέγω, ὅτι τῶν ΑΒΓ, Let ABC and ADE be equal triangles having one an-
ΑΔΕ τριγώνων ἀντιπεπόνθασιν αἱ πλευραὶ αἱ περὶ τὰς ἴσας gle equal to one (angle), (namely) BAC (equal) to DAE.
γωνίας, τουτέστιν, ὅτι ἐστὶν ὡς ἡ ΓΑ πρὸς τὴν ΑΔ, οὕτως I say that, in triangles ABC and ADE, the sides about the

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