Page 176 - Euclid's Elements of Geometry

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ELEMENTS BOOK 6

τῆς πρώτης εἶδος πρὸς τὸ ἀπὸ τῆς δευτέρας τὸ ὅμοιον καὶ (described) on the first (is) to the similar, and similarly
ὁμοίως ἀναγραφόμενον. ὅπερ ἔδει δεῖξαι. described, (figure) on the second. (Which is) the very

† Literally, “double”.

Proposition 20
.
Τὰ ὅμοια πολύγωνα εἴς τε ὅμοια τρίγωνα διαιρεῖται καὶ Similar polygons can be divided into equal numbers
εἰς ἴσα τὸ πλῆθος καὶ ὁμόλογα τοῖς ὅλοις, καὶ τὸ πολύγωνον of similar triangles corresponding (in proportion) to the
πρὸς τὸ πολύγωνον διπλασίονα λόγον ἔχει ἤπερ ἡ ὁμόλογος wholes, and one polygon has to the (other) polygon a
πλευρὰ πρὸς τὴν ὁμόλογον πλευράν. squared ratio with respect to (that) a corresponding side
(has) to a corresponding side.
Α A
Ζ F

Ε E
Β Μ Η Ν Λ B M G N L

Θ Κ H K
Γ ∆ C D
῎Εστω ὅμοια πολύγωνα τὰ ΑΒΓΔΕ, ΖΗΘΚΛ, ὁμόλογος Let ABCDE and FGHKL be similar polygons, and
δὲ ἔστω ἡ ΑΒ τῇ ΖΗ· λέγω, ὅτι τὰ ΑΒΓΔΕ, ΖΗΘΚΛ let AB correspond to FG. I say that polygons ABCDE
πολύγωνα εἴς τε ὅμοια τρίγωνα διαιρεῖται καὶ εἰς ἴσα τὸ and FGHKL can be divided into equal numbers of simi-
πλῆθος καὶ ὁμόλογα τοῖς ὅλοις, καὶ τὸ ΑΒΓΔΕ πολύγωνον lar triangles corresponding (in proportion) to the wholes,
πρὸς τὸ ΖΗΘΚΛ πολύγωνον διπλασίονα λόγον ἔχει ἤπερ ἡ and (that) polygon ABCDE has a squared ratio to poly-
ΑΒ πρὸς τὴν ΖΗ. gon FGHKL with respect to that AB (has) to FG.
᾿Επεζεύχθωσαν αἱ ΒΕ, ΕΓ, ΗΛ, ΛΘ. Let BE, EC, GL, and LH have been joined.
Καὶ ἐπεὶ ὅμοιόν ἐστι τὸ ΑΒΓΔΕ πολύγωνον τῷ And since polygon ABCDE is similar to polygon
ΖΗΘΚΛ πολυγώνῳ, ἴση ἐστὶν ἡ ὑπὸ ΒΑΕ γωνία τῇ ὑπὸ FGHKL, angle BAE is equal to angle GFL, and as BA
ΗΖΛ. καί ἐστιν ὡς ἡ ΒΑ πρὸς ΑΕ, οὕτως ἡ ΗΖ πρὸς ΖΛ. is to AE, so GF (is) to FL [Def. 6.1]. Therefore, since
ἐπεὶ οὖν δύο τρίγωνά ἐστι τὰ ΑΒΕ, ΖΗΛ μίαν γωνίαν μιᾷ ABE and FGL are two triangles having one angle equal
γωνίᾳ ἴσην ἔχοντα, περὶ δὲ τὰς ἴσας γωνίας τὰς πλευρὰς to one angle and the sides about the equal angles propor-
ἀνάλογον, ἰσογώνιον ἄρα ἐστὶ τὸ ΑΒΕ τρίγωνον τῷ ΖΗΛ tional, triangle ABE is thus equiangular to triangle FGL
τριγώνῳ· ὥστε καὶ ὅμοιον· ἴση ἄρα ἐστὶν ἡ ὑπὸ ΑΒΕ γωνία [Prop. 6.6]. Hence, (they are) also similar [Prop. 6.4,
τῇ ὑπὸ ΖΗΛ. ἔστι δὲ καὶ ὅλη ἡ ὑπὸ ΑΒΓ ὅλῃ τῇ ὑπὸ Def. 6.1]. Thus, angle ABE is equal to (angle) FGL.
ΖΗΘ ἴση διὰ τὴν ὁμοιότητα τῶν πολυγώνων· λοιπὴ ἄρα ἡ And the whole (angle) ABC is equal to the whole (angle)
ὑπὸ ΕΒΓ γωνία τῇ ὑπὸ ΛΗΘ ἐστιν ἴση. καὶ ἐπεὶ διὰ τὴν FGH, on account of the similarity of the polygons. Thus,
ὁμοιότητα τῶν ΑΒΕ, ΖΗΛ τριγώνων ἐστὶν ὡς ἡ ΕΒ πρὸς the remaining angle EBC is equal to LGH. And since,
ΒΑ, οὕτως ἡ ΛΗ πρὸς ΗΖ, ἀλλὰ μὴν καὶ διὰ τὴν ὁμοιότητα on account of the similarity of triangles ABE and FGL,
τῶν πολυγώνων ἐστὶν ὡς ἡ ΑΒ πρὸς ΒΓ, οὕτως ἡ ΖΗ πρὸς as EB is to BA, so LG (is) to GF, but also, on account of
ΗΘ, δι᾿ ἴσου ἄρα ἐστὶν ὡς ἡ ΕΒ πρὸς ΒΓ, οὕτως ἡ ΛΗ πρὸς the similarity of the polygons, as AB is to BC, so FG (is)
ΗΘ, καὶ περὶ τὰς ἴσας γωνάις τὰς ὑπὸ ΕΒΓ, ΛΗΘ αἱ πλευραὶ to GH, thus, via equality, as EB is to BC, so LG (is) to
ἀνάλογόν εἰσιν· ἰσογώνιον ἄρα ἐστὶ τὸ ΕΒΓ τρίγωνον τῷ GH [Prop. 5.22], and the sides about the equal angles,
ΛΗΘ τριγώνῳ· ὥστε καὶ ὅμοιόν ἐστι τὸ ΕΒΓ τρίγωνον EBC and LGH, are proportional. Thus, triangle EBC is
τῷ ΛΗΘ τριγώνω. διὰ τὰ αὐτὰ δὴ καὶ τὸ ΕΓΔ τρίγωνον equiangular to triangle LGH [Prop. 6.6]. Hence, triangle
ὅμοιόν ἐστι τῷ ΛΘΚ τριγώνῳ. τὰ ἄρα ὅμοια πολύγωνα τὰ EBC is also similar to triangle LGH [Prop. 6.4, Def. 6.1].
ΑΒΓΔΕ, ΖΗΘΚΛ εἴς τε ὅμοια τρίγωνα διῄρηται καὶ εἰς ἴσα So, for the same (reasons), triangle ECD is also similar

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