Page 75 - Euclid's Elements of Geometry

ST EW gþ.

þ have the same center. (Which is) the very thing it was
ELEMENTS BOOK 3

αὐτῶν τὸ αὐτὸ κέντρον· ὅπερ ἔδει δεῖξαι. the (common) center of the circles ABC and CDG.
Thus, if two circles cut one another then they will not

required to show.

.
Proposition 6
᾿Εὰν δύο κύκλοι ἐφάπτωνται ἀλλήλων, οὐκ ἔσται αὐτῶν If two circles touch one another then they will not
τὸ αὐτὸ κέντρον. have the same center.
Γ C

Ζ F
Ε Β E B

∆ D

Α A
Δύο γὰρ κύκλοι οἱ ΑΒΓ, ΓΔΕ ἐφαπτέσθωσαν ἀλλήλων For let the two circles ABC and CDE touch one an-
κατὰ τὸ Γ σημεῖον· λέγω, ὅτι οὐκ ἔσται αὐτῶν τὸ αὐτὸ other at point C. I say that they will not have the same
κέντρον. center.
Εἰ γὰρ δυνατόν, ἔστω τὸ Ζ, καὶ ἐπεζεύχθω ἡ ΖΓ, καὶ For, if possible, let F be (the common center), and
διήχθω, ὡς ἔτυχεν, ἡ ΖΕΒ. let FC have been joined, and let FEB have been drawn
᾿Επεὶ οὖν τὸ Ζ σημεῖον κέντρον ἐστὶ τοῦ ΑΒΓ κύκλου, through (the two circles), at random.
ἴση ἐστὶν ἡ ΖΓ τῇ ΖΒ. πάλιν, ἐπεὶ τὸ Ζ σημεῖον κέντρον Therefore, since point F is the center of the circle
ἐστὶ τοῦ ΓΔΕ κύκλου, ἴση ἐστὶν ἡ ΖΓ τῇ ΖΕ. ἐδείχθη δὲ ἡ ABC, FC is equal to FB. Again, since point F is the
zþ not have the same center. (Which is) the very thing it was
ΖΓ τῇ ΖΒ ἴση· καὶ ἡ ΖΕ ἄρα τῇ ΖΒ ἐστιν ἴση, ἡ ἐλάττων center of the circle CDE, FC is equal to FE. But FC
τῇ μείζονι· ὅπερ ἐστὶν ἀδύνατον. οὐκ ἄρα τὸ Ζ σημεῖον was shown (to be) equal to FB. Thus, FE is also equal
κέντρον ἐστὶ τῶν ΑΒΓ, ΓΔΕ κύκλων. to FB, the lesser to the greater. The very thing is impos-
᾿Εὰν ἄρα δύο κύκλοι ἐφάπτωνται ἀλλήλων, οὐκ ἔσται sible. Thus, point F is not the (common) center of the
αὐτῶν τὸ αὐτὸ κέντρον· ὅπερ ἔδει δεῖξαι. circles ABC and CDE.
Thus, if two circles touch one another then they will

required to show.

Proposition 7
.
᾿Εὰν κύκλου ἐπὶ τῆς διαμέτρου ληφθῇ τι σημεῖον, ὃ μή If some point, which is not the center of the circle,
ἐστι κέντρον τοῦ κύκλου, ἀπὸ δὲ τοῦ σημείου πρὸς τὸν is taken on the diameter of a circle, and some straight-
κύκλον προσπίπτωσιν εὐθεῖαί τινες, μεγίστη μὲν ἔσται, ἐφ᾿ lines radiate from the point towards the (circumference
ἧς τὸ κέντρον, ἐλαχίστη δὲ ἡ λοιπή, τῶν δὲ ἄλλων ἀεὶ ἡ of the) circle, then the greatest (straight-line) will be that
ἔγγιον τῆς δὶα τοῦ κέντρου τῆς ἀπώτερον μείζων ἐστίν, on which the center (lies), and the least the remainder
δύο δὲ μόνον ἴσαι ἀπὸ τοῦ σημείου προσπεσοῦνται πρὸς (of the same diameter). And for the others, a (straight-
†
τὸν κύκλον ἐφ᾿ ἑκάτερα τῆς ἐλαχίστης. line) nearer to the (straight-line) through the center is
always greater than a (straight-line) further away. And
only two equal (straight-lines) will radiate from the point
towards the (circumference of the) circle, (one) on each

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