Euclid · Fragments §6.20-6.22

Ratio of Triangles and Rectangles with Equal or Supplementary Angles

Passage 15 of 29 · Greek

Summary

Proves geometrically that for two triangles with equal or supplementary angles, the ratio of the rectangles contained by the containing sides equals the ratio of the triangles, and establishes a relation of squares from specific segment ratios.

§6.20κ΄.

20.

Ἔστω δύο τρίγωνα τὰ ΑΒΓ, ∠ΕΖ ἴσας ἔχοντα τὰς Α, ∠ γωνίας· ὅτι ἐστίν, ὡς τὸ ὑπὸ ΒΑΓ πρὸς τὸ ὑπὸ Ε∠Ζ, οὕτως τὸ ΑΒΓ τρίγωνον πρὸς τὸ Ε∠ τρίγωνον.

Let there be two triangles ΑΒΓ, ∠ΕΖ having the angles Α, ∠ equal; (to prove) that, as the rectangle contained by ΒΑ, ΑΓ is to the rectangle contained by Ε∠, ∠Ζ, so is the triangle ΑΒΓ to the triangle ∠ΕΖ.

ἤχθωσαν κάθετοι αἱ ΒΗ, ΕΘ. ἐπεὶ οὖν ἴση ἐστὶν ἡ μὲν Α γωνία τῇ ∠, ἡ δὲ Η τῇ Θ, ἔστιν ἄρα, ὡς ἡ ΑΒ πρὸς τὴν ΒΗ, οὕτως ἡ ∠Ε πρὸς τὴν ΕΘ. ἀλλ᾿ ὡς μὲν ἡ ΑΒ πρὸς τὴν ΒΗ, οὕτως ἐστὶν τὸ ὑπὸ ΒΑΓ πρὸς τὸ ὑπὸ ΒΗ, ΑΓ, ὡς δὲ ἡ ∠Ε πρὸς τὴν ΕΘ, οὕτως ἐστὶν τὸ ὑπὸ Ε∠Ζ πρὸς τὸ ὑπὸ ΕΘ, ∠Ζ ἔστιν ἄρα, ὡς τὸ ὑπὸ ΒΑΓ πρὸς τὸ ὑπὸ ΒΗ, ΑΓ, οὕτως τὸ ὑπὸ Ε∠Ζ πρὸς τὸ ὑπὸ ΕΘ, ∠Ζ· καὶ ἐναλλάξ.

Let perpendiculars ΒΗ, ΕΘ be drawn. Since therefore the angle Α is equal to ∠, and Η to Θ, it follows that, as ΑΒ is to ΒΗ, so is ∠Ε to ΕΘ. But as ΑΒ is to ΒΗ, so is the rectangle contained by ΒΑ, ΑΓ to the rectangle contained by ΒΗ, ΑΓ, and as ∠Ε is to ΕΘ, so is the rectangle contained by Ε∠, ∠Ζ to the rectangle contained by ΕΘ, ∠Ζ; therefore, as the rectangle contained by ΒΑ, ΑΓ is to the rectangle contained by ΒΗ, ΑΓ, so is the rectangle contained by Ε∠, ∠Ζ to the rectangle contained by ΕΘ, ∠Ζ; and alternately.

ἀλλ᾿ ὡς τὸ ὑπὸ ΒΗ, ΑΓ πρὸς τὸ ὑπὸ ΕΘ, ∠Ζ, οὕτως ἐστὶν τὸ ΑΒΓ τρίγωνον πρὸς τὸ ∠ΕΖ τρίγωνον· ἐκατέρα γὰρ τῶν ΒΗ, ΕΘ κάθετός ἐστιν ἑκατέρου τῶν εἰρημένων τριγώνων· καὶ ὡς ἄρα τὸ ὑπὸ ΒΑΓ πρὸς τὸ ὑπὸ Ε∠Ζ, οὕτως ἐστὶν τὸ ΑΒΓ τρίγωνον πρὸς τὸ ∠ΕΖ τρίγωνον.

But as the rectangle contained by ΒΗ, ΑΓ is to the rectangle contained by ΕΘ, ∠Ζ, so is the triangle ΑΒΓ to the triangle ∠ΕΖ; for each of ΒΗ, ΕΘ is a perpendicular of each of the said triangles; therefore also, as the rectangle contained by ΒΑ, ΑΓ is to the rectangle contained by Ε∠, ∠Ζ, so is the triangle ΑΒΓ to the triangle ∠ΕΖ.

§6.21κα΄.

21.

Ἔστωσαν δὴ αἱ Α, ∠ δυσὶν ὀρθαῖς ἴσαι· ὅτι πάλιν γίνεται, ὡς τὸ ὑπὸ ΒΑΓ πρὸς τὸ ὑπὸ Ε∠Ζ, οὕτως τὸ ΑΒΓ τρίγωνον πρὸς τὸ ∠ΕΖ τρίγωνον.

Now let the angles Α, ∠ be equal to two right angles; (to prove) that again, as the rectangle contained by ΒΑ, ΑΓ is to the rectangle contained by Ε∠, ∠Ζ, so is the triangle ΑΒΓ to the triangle ∠ΕΖ.

ἐκβεβλήσθω ἡ ΒΑ, καὶ κείσθω τῇ ΒΑ ἴση ἡ ΑΗ, καὶ ἐπεζεύχθω ἡ ΓΗ. ἐπεὶ οὖν αἱ Α, Δ γωνίαι δυσὶν ὀρθαῖς ἴσαι εἰσίν, ἀλλὰ καὶ αἱ ὑπὸ ΒΑΓ, ΓΑΗ γωνίαι δυσὶν ὀρθαῖς, ἴση ἄρα ἐστὶν ἡ ὑπὸ ΓΑΗ γωνία τῇ ∠.

Let ΒΑ be produced, and let ΑΗ be placed equal to ΒΑ, and let ΓΗ be joined. Since therefore the angles Α, Δ are equal to two right angles, and also the angles ΒΑΓ, ΓΑΗ are equal to two right angles, therefore the angle ΓΑΗ is equal to ∠.

ἔστιν οὖν, ὡς τὸ ὑπὸ ΗΑΓ πρὸς τὸ ὑπὸ Ε∠Ζ, οὕτως τὸ ΑΗΓ τρίγωνον πρὸς τὸ ∠ΕΖ τρίγωνον ἴση δέ ἐστιν ἡ μὲν ΗΑ τῇ ΑΒ, τὸ δὲ ΗΑΓ τρίγωνον τῷ ΑΒΓ τριγώνῳ· ἔστιν ἄρα, ὡς τὸ ὑπὸ ΒΑΓ πρὸς τὸ ὑπὸ Ε∠Ζ, οὕτως τὸ ΑΒΓ τρίγωνον πρὸς τὸ ∠ΕΖ τρίγωνον.

Therefore, as the rectangle contained by ΗΑ, ΑΓ is to the rectangle contained by Ε∠, ∠Ζ, so is the triangle ΑΗΓ to the triangle ∠ΕΖ. And ΗΑ is equal to ΑΒ, and the triangle ΑΗΓ is equal to the triangle ΑΒΓ; therefore, as the rectangle contained by ΒΑ, ΑΓ is to the rectangle contained by Ε∠, ∠Ζ, so is the triangle ΑΒΓ to the triangle ∠ΕΖ.

§6.22κβ΄.

22.

Εὐθεῖα ἡ ΑΒ, καὶ ἐπ᾿ αὐτῆς δύο σημεῖα τὰ Γ, ∠, ἔστω δὲ τὸ δὶς ὑπὸ ΑΒ, Γ∠ ἴσον τῷ ἀπὸ ΓΒ· ὅτι καὶ τὸ ἀπὸ Α∠ ἴσον ἐστὶν τοῖς ἀπὸ τῶν Α ∠Β τετραγώνοις.

Let there be a straight line ΑΒ, and on it two points Γ, ∠, and let twice the rectangle contained by ΑΒ, Γ∠ be equal to the square on ΓΒ; (to prove) that the square on Α∠ is also equal to the squares on ΑΓ, ∠Β.

ἐπεὶ γὰρ τὸ δὶς ὑπὸ ΑΒ, Γγώνοις. ἴσον ἐστὶ τῷ ἀπὸ ΓΒ, κοινὸν ἀφῃρήσθω τὸ δὶς ὑπὸ Β∠Γ λοιπὸν ἄρα τὸ δὲς ὑπὸ Α∠Γ ἴσον ἐστὶν τοῖς ἀπὸ τῶν Γ∠, ∠Β τετραγώνοις.

For since twice the rectangle contained by ΑΒ, Γ∠ is equal to the square on ΓΒ, let the common twice the rectangle contained by Β∠, Γ be subtracted; therefore, the remaining twice the rectangle contained by Α∠, Γ is equal to the squares on Γ∠, ∠Β.

κοινὸν ἀφῃρήσθω τὸ ἀπὸ ΓΔ τετράγωνον· λοιπὸν ἄρα τὸ θὶς ὑπὸ ΑΓ∠ μετὰ τοῦ ἀπὸ Γ∠ ἴσον ἐστὶν τῷ ἀπὸ ∠Β τετραγώνῳ.

Let the common square on ΓΔ be subtracted; therefore, the remaining twice the rectangle contained by ΑΓ, ∠ together with the square on Γ∠ is equal to the square on ∠Β.

κοινὸν προσκείσθω τὸ ἀπὸ ΑΓ τετράγωνον· ὅλον ἄρα τὸ ἀπὸ Α∠ τετράγωνον ἴσον ἐστὶν τοῖς ἀπὸ τῶν ΑΓ, ∠Β τετραγώνοις.

Let the common square on ΑΓ be added; therefore, the whole square on Α∠ is equal to the squares on ΑΓ, ∠Β.

Notes

§6.20τὸ ὑπὸ ΒΑΓ — A conventional geometrical formula meaning 'the rectangle contained by BA and AΓ'. The neuter noun 'rectangle' (χωρίον), agreeing with the article τό, is omitted, and the preposition ὑπό designates containment under the two sides.
§6.21δυσὶν ὀρθαῖς ἴσαι — Meaning 'equal to two right angles' (180 degrees). Since the adjective ἴσαι is feminine plural nominative, the omitted subject is understood as 'angles' (γωνίαι), and 'two' (δυσίν) 'right' (ὀρθαῖς) function as dative of reference.
§6.22Γγώνοις. — A textual corruption in the manuscript. Judging from the mathematical consistency and the immediate context, the text originally read `Γ∠ ἴσον`, which was corrupted into `Γγώνοις. ἴσον` due to graphic similarity. The translation corrects this to mean 'twice Γ∠ is equal to'.
§6.22ΓΔ — Since the two points on the line are designated as Γ and ∠, this mathematically refers to `Γ∠` (the square on Γ∠). In the manuscript tradition, the symbol ∠ (representing Delta) was likely conflated with the standard letter Δ (Delta), resulting in `ΓΔ`. The translation preserves the textual representation while understanding it geometrically as Γ∠.

Cite this passage

Euclid, Fragments §6.20-6.22. Humanitext Reader, https://reader.humanitext.ai/en/text/urn:cts:greekLit:tlg1799.tlg016.humanitext-grc1:6.20-6.22

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