§3.prop.6ἐὰν δύο κύκλοι ἐφάπτωνται ἀλλήλων, οὐκ ἔσται αὐτῶν τὸ αὐτὸ κέντρον.
If two circles touch one another, they will not have the same center.
δύο γὰρ κύκλοι οἱ ΑΒΓ, ΓΔΕ ἐφαπτέσθωσαν ἀλλήλων κατὰ τὸ Γ σημεῖον· λέγω, ὅτι οὐκ ἔσται αὐτῶν τὸ αὐτὸ κέντρον.
For let two circles ΑΒΓ, ΓΔΕ touch one another at the point Γ; I say that they will not have the same center.
εἰ γὰρ δυνατόν, ἔστω τὸ Ζ, καὶ ἐπεζεύχθω ἡ ΖΓ, καὶ διήχθω, ὡς ἔτυχεν, ἡ ΖΕΒ.
ἐπεὶ οὖν τὸ Ζ σημεῖον κέντρον ἐστὶ τοῦ ΑΒΓ κύκλου, ἴση ἐστὶν ἡ ΖΓ τῇ ΖΒ. πάλιν, ἐπεὶ τὸ Ζ σημεῖον κέντρον ἐστὶ τοῦ ΓΔΕ κύκλου, ἴση ἐστὶν ἡ ΖΓ τῇ ΖΕ. ἐδείχθη δὲ ἡ ΖΓ τῇ ΖΒ ἴση· καὶ ἡ ΖΕ ἄρα τῇ ΖΒ ἐστιν ἴση, ἡ ἐλάττων τῇ μείζονι·
For, if possible, let it be Ζ, and let ΖΓ be joined, and let ΖΕΒ be drawn at random. Since then the point Ζ is the center of the circle ΑΒΓ, ΖΓ is equal to ΖΒ. Again, since the point Ζ is the center of the circle ΓΔΕ, ΖΓ is equal to ΖΕ. But ΖΓ was also proved equal to ΖΒ; therefore ΖΕ is also equal to ΖΒ, the less to the greater: which is impossible.
ὅπερ ἐστὶν ἀδύνατον. οὐκ ἄρα τὸ Ζ σημεῖον κέντρον ἐστὶ τῶν ΑΒΓ, ΓΔΕ κύκλων.
Therefore the point Ζ is not the center of the circles ΑΒΓ, ΓΔΕ.
ἐὰν ἄρα δύο κύκλοι ἐφάπτωνται ἀλλήλων, οὐκ ἔσται αὐτῶν τὸ αὐτὸ κέντρον· ὅπερ ἔδει δεῖξαι.
If then two circles touch one another, they will not have the same center; which was meet to show.
§3.prop.7ἐὰν κύκλου ἐπὶ τῆς διαμέτρου ληφθῇ τι σημεῖον, ὃ μή ἐστι κέντρον τοῦ κύκλου, ἀπὸ δὲ τοῦ σημείου πρὸς τὸν κύκλον προσπίπτωσιν εὐθεῖαί τινες, μεγίστη μὲν ἔσται, ἐφʼ ἧς τὸ κέντρον, ἐλαχίστη δὲ ἡ λοιπή, τῶν δὲ ἄλλων ἀεὶ ἡ ἔγγιον τῆς διὰ τοῦ κέντρου τῆς ἀπώτερον μείζων ἐστίν, δύο δὲ μόνον ἴσαι ἀπὸ τοῦ σημείου προσπεσοῦνται πρὸς τὸν κύκλον ἐφʼ ἑκάτερα τῆς ἐλαχίστης.
If on the diameter of a circle some point be taken which is not the center of the circle, and from the point some straight lines fall on the circle, the greatest will be that on which the center is, and the least the remainder, while of the others the nearer to the one through the center is always greater than the more remote, and only two equal straight lines will fall from the point on the circle on either side of the least.
ἔστω κύκλος ὁ ΑΒΓΔ, διάμετρος δὲ αὐτοῦ ἔστω ἡ ΑΔ, καὶ ἐπὶ τῆς ΑΔ εἰλήφθω τι σημεῖον τὸ Ζ, ὃ μή ἐστι κέντρον τοῦ κύκλου, κέντρον δὲ τοῦ κύκλου ἔστω τὸ Ε, καὶ ἀπὸ τοῦ Ζ πρὸς τὸν ΑΒΓΔ κύκλον προσπιπτέτωσαν εὐθεῖαί τινες αἱ ΖΒ, ΖΓ, ΖΗ· λέγω, ὅτι μεγίστη μέν ἐστιν ἡ ΖΑ, ἐλαχίστη δὲ ἡ ΖΔ, τῶν δὲ ἄλλων ἡ μὲν ΖΒ τῆς ΖΓ μείζων, ἡ δὲ ΖΓ τῆς ΖΗ.
ἐπεζεύχθωσαν γὰρ αἱ ΒΕ, ΓΕ, ΗΕ. καὶ ἐπεὶ παντὸς τριγώνου αἱ δύο πλευραὶ τῆς λοιπῆς μείζονές εἰσιν, αἱ ἄρα ΕΒ, ΕΖ τῆς ΒΖ μείζονές εἰσιν.
Let ΑΒΓΔ be a circle, and let ΑΔ be its diameter, and on ΑΔ let some point Ζ be taken which is not the center of the circle, and let Ε be the center of the circle, and from Ζ let some straight lines ΖΒ, ΖΓ, ΖΗ fall on the circle ΑΒΓΔ; I say that ΖΑ is the greatest, and ΖΔ the least, and of the others ΖΒ is greater than ΖΓ, and ΖΓ than ΖΗ. For let ΒΕ, ΓΕ, ΗΕ be joined. And since in any triangle two sides are greater than the remaining one, therefore ΕΒ, ΕΖ are greater than ΒΖ.
ἴση δὲ ἡ ΑΕ τῇ ΒΕ· μείζων ἄρα ἡ ΑΖ τῆς ΒΖ. πάλιν, ἐπεὶ ἴση ἐστὶν ἡ ΒΕ τῇ ΓΕ, κοινὴ δὲ ἡ ΖΕ, δύο δὴ αἱ ΒΕ, ΕΖ δυσὶ ταῖς ΓΕ, ΕΖ ἴσαι εἰσίν.
And ΑΕ is equal to ΒΕ; therefore ΑΖ is greater than ΒΖ. Again, since ΒΕ is equal to ΓΕ, and ΖΕ is common, the two ΒΕ, ΕΖ are equal to the two ΓΕ, ΕΖ.
ἀλλὰ καὶ γωνία ἡ ὑπὸ ΒΕΖ γωνίας τῆς ὑπὸ ΓΕΖ μείζων. βάσις ἄρα ἡ ΒΖ βάσεως τῆς ΓΖ μείζων ἐστίν.
But the angle ΒΕΖ is also greater than the angle ΓΕΖ; therefore the base ΒΖ is greater than the base ΓΖ.
διὰ τὰ αὐτὰ δὴ καὶ ἡ ΓΖ τῆς ΖΗ μείζων ἐστίν.
For the same reasons, ΓΖ is also greater than ΖΗ.
πάλιν, ἐπεὶ αἱ ΗΖ, ΖΕ τῆς ΕΗ μείζονές εἰσιν, ἴση δὲ ἡ ΕΗ τῇ ΕΔ, αἱ ἄρα ΗΖ, ΖΕ τῆς ΕΔ μείζονές εἰσιν.
Again, since ΗΖ, ΖΕ are greater than ΕΗ, and ΕΗ is equal to ΕΔ, therefore ΗΖ, ΖΕ are greater than ΕΔ.
κοινὴ ἀφῃρήσθω ἡ ΕΖ· λοιπὴ ἄρα ἡ ΗΖ λοιπῆς τῆς ΖΔ μείζων ἐστίν.
Let the common ΕΖ be subtracted; therefore the remainder ΗΖ is greater than the remainder ΖΔ.
μεγίστη μὲν ἄρα ἡ ΖΑ, ἐλαχίστη δὲ ἡ ΖΔ, μείζων δὲ ἡ μὲν ΖΒ τῆς ΖΓ, ἡ δὲ ΖΓ τῆς ΖΗ.
λέγω, ὅτι καὶ ἀπὸ τοῦ Ζ σημείου δύο μόνον ἴσαι προσπεσοῦνται πρὸς τὸν ΑΒΓΔ κύκλον ἐφʼ ἑκάτερα τῆς ΖΔ ἐλαχίστης.
Therefore ΖΑ is the greatest, and ΖΔ the least, and ΖΒ is greater than ΖΓ, and ΖΓ than ΖΗ. I say also that from the point Ζ only two equal straight lines will fall on the circle ΑΒΓΔ on either side of the least ΖΔ.
συνεστάτω γὰρ πρὸς τῇ ΕΖ εὐθείᾳ καὶ τῷ πρὸς αὐτῇ σημείῳ τῷ Ε τῇ ὑπὸ ΗΕΖ γωνίᾳ ἴση ἡ ὑπὸ ΖΕΘ, καὶ ἐπεζεύχθω ἡ ΖΘ. ἐπεὶ οὖν ἴση ἐστὶν ἡ ΗΕ τῇ ΕΘ, κοινὴ δὲ ἡ ΕΖ, δύο δὴ αἱ ΗΕ, ΕΖ δυσὶ ταῖς ΘΕ, ΕΖ ἴσαι εἰσίν· καὶ γωνία ἡ ὑπὸ ΗΕΖ γωνίᾳ τῇ ὑπὸ ΘΕΖ ἴση· βάσις ἄρα ἡ ΖΗ βάσει τῇ ΖΘ ἴση ἐστίν.
For let the angle ΖΕΘ be constructed equal to the angle ΗΕΖ on the straight line ΕΖ and at the point Ε on it, and let ΖΘ be joined. Since then ΗΕ is equal to ΕΘ, and ΕΖ is common, the two ΗΕ, ΕΖ are equal to the two ΘΕ, ΕΖ; and the angle ΗΕΖ is equal to the angle ΘΕΖ; therefore the base ΖΗ is equal to the base ΖΘ.
λέγω δή, ὅτι τῇ ΖΗ ἄλλη ἴση οὐ προσπεσεῖται πρὸς τὸν κύκλον ἀπὸ τοῦ Ζ σημείου.
I say then that another straight line equal to ΖΗ will not fall on the circle from the point Ζ.
εἰ γὰρ δυνατόν, προσπιπτέτω ἡ ΖΚ. καὶ ἐπεὶ ἡ ΖΚ τῇ ΖΗ ἴση ἐστίν, ἀλλὰ ἡ ΖΘ τῇ ΖΗ, καὶ ἡ ΖΚ ἄρα τῇ ΖΘ ἐστιν ἴση, ἡ ἔγγιον τῆς διὰ τοῦ κέντρου τῇ ἀπώτερον ἴση· ὅπερ ἀδύνατον.
For, if possible, let ΖΚ fall. And since ΖΚ is equal to ΖΗ, but ΖΘ is equal to ΖΗ, therefore ΖΚ is also equal to ΖΘ, the nearer to the one through the center equal to the more remote: which is impossible.
οὐκ ἄρα ἀπὸ τοῦ Ζ σημείου ἑτέρα τις προσπεσεῖται πρὸς τὸν κύκλον ἴση τῇ ΗΖ· μία ἄρα μόνη.
Therefore another straight line equal to ΗΖ will not fall from the point Ζ on the circle; therefore only one.
ἐὰν ἄρα κύκλου ἐπὶ τῆς διαμέτρου ληφθῇ τι σημεῖον, ὃ μή ἐστι κέντρον τοῦ κύκλου, ἀπὸ δὲ τοῦ σημείου πρὸς τὸν κύκλον προσπίπτωσιν εὐθεῖαί τινες, μεγίστη μὲν ἔσται, ἐφʼ ἧς τὸ κέντρον, ἐλαχίστη δὲ ἡ λοιπή, τῶν δὲ ἄλλων ἀεὶ ἡ ἔγγιον τῆς διὰ τοῦ κέντρου τῆς ἀπώτερον μείζων ἐστίν, δύο δὲ μόνον ἴσαι ἀπὸ τοῦ αὐτοῦ σημείου προσπεσοῦνται πρὸς τὸν κύκλον ἐφʼ ἑκάτερα τῆς ἐλαχίστης· ὅπερ ἔδει δεῖξαι.
If then on the diameter of a circle some point be taken which is not the center of the circle, and from the point some straight lines fall on the circle, the greatest will be that on which the center is, and the least the remainder, while of the others the nearer to the one through the center is always greater than the more remote, and only two equal straight lines will fall from the same point on the circle on either side of the least; which was meet to show.