Page 126 - Euclid's Elements of Geometry

ST EW dþ.

ELEMENTS BOOK 4

καὶ ἰσογώνιον ἐγγέγραπται· ὅπερ ἔδει ποιῆσαι. to DEF [Prop. 3.27]. Similarly, it can also be shown
ìri sma Thus, hexagon ABCDEF is equiangular. And it was also
that the remaining angles of hexagon ABCDEF are in-
dividually equal to each of the angles AFE and FED.

shown (to be) equilateral. And it has been inscribed in
circle ABCDE.
Thus, an equilateral and equiangular hexagon has
been inscribed in the given circle. (Which is) the very
thing it was required to do.
.
Corollary
᾿Εκ δὴ τούτου φανερόν, ὅτι ἡ τοῦ ἑξαγώνου πλευρὰ ἴση So, from this, (it is) manifest that a side of the
ἐστὶ τῇ ἐκ τοῦ κέντρου τοῦ κύκλου. hexagon is equal to the radius of the circle.
῾Ομοίως δὲ τοῖς ἐπὶ τοῦ πενταγώνου ἐὰν διὰ τῶν κατὰ And similarly to a pentagon, if we draw tangents
τὸν κύκλον διαιρέσεων ἐφαπτομένας τοῦ κύκλου ἀγάγωμεν, to the circle through the (sixfold) divisions of the (cir-
iþ hexagon. (Which is) the very thing it was required to do.
περιγραφήσεται περὶ τὸν κύκλον ἑξάγωνον ἰσόπλευρόν cumference of the) circle, an equilateral and equiangu-
τε καὶ ἰσογώνιον ἀκολούθως τοῖς ἐπὶ τοῦ πενταγώνου lar hexagon can be circumscribed about the circle, analo-
εἰρημένοις. καὶ ἔτι διὰ τῶν ὁμοίων τοῖς ἐπὶ τοῦ πενταγώνου gously to the aforementioned pentagon. And, further, by
εἰρημένοις εἰς τὸ δοθὲν ἑξάγωνον κύκλον ἐγγράψομέν τε (means) similar to the aforementioned pentagon, we can
καὶ περιγράψομεν· ὅπερ ἔδει ποιῆσαι. inscribe and circumscribe a circle in (and about) a given

† See the footnote to Prop. 4.6.

Proposition 16
.
Εἰς τὸν δοθέντα κύκλον πεντεκαιδεκάγωνον ἰσόπλευρόν To inscribe an equilateral and equiangular fifteen-
τε καὶ ἰσογώνιον ἐγγράψαι. sided figure in a given circle.
Α A

Β B

Ε E

Γ ∆ C D

῎Εστω ὁ δοθεὶς κύκλος ὁ ΑΒΓΔ· δεῖ δὴ εἰς τὸν ΑΒΓΔ Let ABCD be the given circle. So it is required to in-
κύκλον πεντεκαιδεκάγωνον ἰσόπλευρόν τε καὶ ἰσογώνιον scribe an equilateral and equiangular fifteen-sided figure
ἐγγράψαι. in circle ABCD.
᾿Εγγεγράφθω εἰς τὸν ΑΒΓΔ κύκλον τριγώνου μὲν ἰσο- Let the side AC of an equilateral triangle inscribed
πλεύρου τοῦ εἰς αὐτὸν ἐγγραφομένου πλευρὰ ἡ ΑΓ, πεν- in (the circle) [Prop. 4.2], and (the side) AB of an (in-
ταγώνου δὲ ἰσοπλεύρου ἡ ΑΒ· οἵων ἄρα ἐστὶν ὁ ΑΒΓΔ scribed) equilateral pentagon [Prop. 4.11], have been in-
κύκλος ἴσων τμήματων δεκαπέντε, τοιούτων ἡ μὲν ΑΒΓ scribed in circle ABCD. Thus, just as the circle ABCD
περιφέρεια τρίτον οὖσα τοῦ κύκλου ἔσται πέντε, ἡ δὲ ΑΒ is (made up) of fifteen equal pieces, the circumference
περιφέρεια πέμτον οὖσα τοῦ κύκλου ἔσται τριῶν· λοιπὴ ἄρα ABC, being a third of the circle, will be (made up) of five

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