Humanitext Reader

Euclid · Elements §13.prop.12

Ratio of Square on Inscribed Triangle Side to Radius

Passage 305 of 316 · Greek

Summary

This proposition proves that the square on the side of an equilateral triangle inscribed in a circle is triple the square on the radius of the circle, by utilizing the extension of the diameter and the right angle in a semicircle.

§13.prop.12ἐὰν εἰς κύκλον τρίγωνον ἰσόπλευρον ἐγγραφῇ, ἡ τοῦ τριγώνου πλευρὰ δυνάμει τριπλασίων ἐστὶ τῆς ἐκ τοῦ κέντρου τοῦ κύκλου.
If an equilateral triangle be inscribed in a circle, the side of the triangle is triple in square of the radius of the circle.
ἔστω κύκλος ὁ ΑΒΓ, καὶ εἰς αὐτὸν τρίγωνον ἰσόπλευρον ἐγγεγράφθω τὸ ΑΒΓ· λέγω, ὅτι τοῦ ΑΒΓ τριγώνου μία πλευρὰ δυνάμει τριπλασίων ἐστὶ τῆς ἐκ τοῦ κέντρου τοῦ ΑΒΓ κύκλου.
Let the circle be ABG, and let the equilateral triangle ABG be inscribed in it; I say that one side of the triangle ABG is triple in square of the radius of the circle ABG.
εἰλήφθω γὰρ τὸ κέντρον τοῦ ΑΒΓ κύκλου τὸ Δ, καὶ ἐπιζευχθεῖσα ἡ ΑΔ διήχθω ἐπὶ τὸ Ε, καὶ ἐπεζεύχθω ἡ ΒΕ. καὶ ἐπεὶ ἰσόπλευρόν ἐστι τὸ ΑΒΓ τρίγωνον, ἡ ΒΕΓ ἄρα περιφέρεια τρίτον μέρος ἐστὶ τῆς τοῦ ΑΒΓ κύκλου περιφερείας.
For let the center of the circle ABG be taken, namely D, and let AD be joined and drawn through to E, and let BE be joined. And since the triangle ABG is equilateral, therefore the circumference BEG is a third part of the circumference of the circle ABG.
ἡ ἄρα ΒΕ περιφέρεια ἕκτον ἐστὶ μέρος τῆς τοῦ κύκλου περιφερείας· ἑξαγώνου ἄρα ἐστὶν ἡ ΒΕ εὐθεῖα· ἴση ἄρα ἐστὶ τῇ ἐκ τοῦ κέντρου τῇ ΔΕ. καὶ ἐπεὶ διπλῆ ἐστιν ἡ ΑΕ τῆς ΔΕ, τετραπλάσιόν ἐστι τὸ ἀπὸ τῆς ΑΕ τοῦ ἀπὸ τῆς ΕΔ, τουτέστι τοῦ ἀπὸ τῆς ΒΕ. ἴσον δὲ τὸ ἀπὸ τῆς ΑΕ τοῖς ἀπὸ τῶν ΑΒ, ΒΕ· τὰ ἄρα ἀπὸ τῶν ΑΒ, ΒΕ τετραπλάσιά ἐστι τοῦ ἀπὸ τῆς ΒΕ. διελόντι ἄρα τὸ ἀπὸ τῆς ΑΒ τριπλάσιόν ἐστι τοῦ ἀπὸ ΒΕ. ἴση δὲ ἡ ΒΕ τῇ ΔΕ· τὸ ἄρα ἀπὸ τῆς ΑΒ τριπλάσιόν ἐστι τοῦ ἀπὸ τῆς ΔΕ. ἡ ἄρα τοῦ τριγώνου πλευρὰ δυνάμει τριπλασία ἐστὶ τῆς ἐκ τοῦ κέντρου· ὅπερ ἔδει δεῖξαι.
Therefore the circumference BE is a sixth part of the circumference of the circle; therefore the straight line BE is the side of a hexagon; therefore it is equal to the radius DE. And since AE is double of DE, the square on AE is quadruple of the square on ED, that is, of the square on BE. But the square on AE is equal to the squares on AB, BE; therefore the squares on AB, BE are quadruple of the square on BE. Therefore, by separation, the square on AB is triple of the square on BE. And BE is equal to DE; therefore the square on AB is triple of the square on DE. Therefore the side of the triangle is triple in square of the radius; which was to be proved.

Notes

  1. §13.prop.12δυνάμει τριπλασίων — The dative δυνάμει (literally 'in power', but in mathematical contexts 'in square') is used with the multiple adjective τριπλασίων ('threefold' or 'triple') to mean that the square of the line is three times the square of the other line.
  2. §13.prop.12τῆς ἐκ τοῦ κέντρου — The feminine article τῆς agrees with the omitted noun εὐθείας ('straight line'), and with the prepositional phrase ἐκ τοῦ κέντρου ('from the center') it means 'the straight line from the center', i.e., the radius. The genitive case is used because it is governed by the multiple adjective τριπλασίων (genitive of comparison).
  3. §13.prop.12διελόντι — The second aorist active participle (masculine dative singular) of διαιρέω ('to divide' or 'to separate'). It is a conventional dative of interest/reference used in Greek mathematics to mean 'by separation' (dividendo) or 'if we subtract (the common part)'. Here it refers to subtracting $BE^2$ from both sides of the equation $AB^2 + BE^2 = 4BE^2$ to obtain $AB^2 = 3BE^2$.
  4. §13.prop.12ἴσον δὲ τὸ ἀπὸ τῆς ΑΕ τοῖς ἀπὸ τῶν ΑΒ, ΒΕ — Since AE is a diameter (passing through the center D), the angle ABE is a right angle because it is in a semicircle (Elements, III. 31). Therefore, by the Pythagorean theorem (Elements, I. 47) applied to the right-angled triangle ABE, the square on AE is equal to the sum of the squares on AB and BE. This geometric fact is introduced directly without explicit reference.

Cite this passage

Euclid, Elements §13.prop.12. Humanitext Reader, https://reader.humanitext.ai/en/text/urn:cts:greekLit:tlg1799.tlg001.humanitext-grc2:13.prop.12

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